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betacdf


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 -- statistics: P = betacdf (X, A, B)
 -- statistics: P = betacdf (X, A, B, "upper")

     Beta cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function of the
     Beta distribution with shape parameters A and B.  The size of P is the
     common size of X, A, and B.  A scalar input functions as a constant matrix
     of the same size as the other inputs.

     ‘P = betacdf (X, A, B, "upper")’ computes the upper tail probability of the
     Beta distribution with parameters A and B, at the values in X.

     Further information about the Beta distribution can be found at
     <https://en.wikipedia.org/wiki/Beta_distribution>

     See also: betainv, betapdf, betarnd, betafit, betalike, betastat.


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Beta cumulative distribution function (CDF).



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betainv


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 -- statistics: X = betainv (P, A, B)

     Inverse of the Beta distribution (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     Beta distribution with shape parameters A and B.  The size of X is the
     common size of X, A, and B.  A scalar input functions as a constant matrix
     of the same size as the other inputs.

     Further information about the Beta distribution can be found at
     <https://en.wikipedia.org/wiki/Beta_distribution>

     See also: betacdf, betapdf, betarnd, betafit, betalike, betastat.


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Inverse of the Beta distribution (iCDF).



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betapdf


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 -- statistics: Y = betapdf (X, A, B)

     Beta probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the Beta distribution with shape parameters A and B.  The size of Y is the
     common size of X, A, and B.  A scalar input functions as a constant matrix
     of the same size as the other inputs.

     Further information about the Beta distribution can be found at
     <https://en.wikipedia.org/wiki/Beta_distribution>

     See also: betacdf, betainv, betarnd, betafit, betalike, betastat.


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Beta probability density function (PDF).



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betarnd


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 -- statistics: R = betarnd (A, B)
 -- statistics: R = betarnd (A, B, ROWS)
 -- statistics: R = betarnd (A, B, ROWS, COLS, ...)
 -- statistics: R = betarnd (A, B, [SZ])

     Random arrays from the Beta distribution.

     ‘R = betarnd (A, B)’ returns an array of random numbers chosen from the
     Beta distribution with shape parameters A and B.  The size of R is the
     common size of A and B.  A scalar input functions as a constant matrix of
     the same size as the other inputs.

     When called with a single size argument, ‘betarnd’ returns a square matrix
     with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     Further information about the Beta distribution can be found at
     <https://en.wikipedia.org/wiki/Beta_distribution>

     See also: betacdf, betainv, betapdf, betafit, betalike, betastat.


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Random arrays from the Beta distribution.



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binocdf


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 -- statistics: P = binocdf (X, N, PS)
 -- statistics: P = binocdf (X, N, PS, "upper")

     Binomial cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the binomial distribution with parameters N and PS, where N is the
     number of trials and PS is the probability of success.  The size of P is
     the common size of X, N, and PS.  A scalar input functions as a constant
     matrix of the same size as the other inputs.

     ‘P = binocdf (X, N, PS, "upper")’ computes the upper tail probability of
     the binomial distribution with parameters N and PS, at the values in X.

     Further information about the binomial distribution can be found at
     <https://en.wikipedia.org/wiki/Binomial_distribution>

     See also: binoinv, binopdf, binornd, binofit, binolike, binostat, binotest.


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Binomial cumulative distribution function (CDF).



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binoinv


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 -- statistics: X = binoinv (P, N, PS)

     Inverse of the Binomial cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     binomial distribution with parameters N and PS, where N is the number of
     trials and PS is the probability of success.  The size of X is the common
     size of P, N, and PS.  A scalar input functions as a constant matrix of the
     same size as the other inputs.

     Further information about the binomial distribution can be found at
     <https://en.wikipedia.org/wiki/Binomial_distribution>

     See also: binocdf, binopdf, binornd, binofit, binolike, binostat, binotest.


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Inverse of the Binomial cumulative distribution function (iCDF).



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binopdf


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 -- statistics: Y = binopdf (X, N, PS)

     Binomial probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the binomial distribution with parameters N and PS, where N is the number
     of trials and PS is the probability of success.  The size of Y is the
     common size of X, N, and PS.  A scalar input functions as a constant matrix
     of the same size as the other inputs.

     Matlab incompatibility: Octave's ‘binopdf’ does not allow complex input
     values.  Matlab 2021b returns values for complex inputs despite the
     documentation indicates integer and real value inputs are required.

     Further information about the binomial distribution can be found at
     <https://en.wikipedia.org/wiki/Binomial_distribution>

     See also: binocdf, binoinv, binornd, binofit, binolike, binostat, binotest.


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Binomial probability density function (PDF).



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binornd


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 -- statistics: R = binornd (N, PS)
 -- statistics: R = binornd (N, PS, ROWS)
 -- statistics: R = binornd (N, PS, ROWS, COLS, ...)
 -- statistics: R = binornd (N, PS, [SZ])

     Random arrays from the Binomial distribution.

     ‘R = binornd (N, PS)’ returns a matrix of random samples from the binomial
     distribution with parameters N and PS, where N is the number of trials and
     PS is the probability of success.  The size of R is the common size of N
     and PS.  A scalar input functions as a constant matrix of the same size as
     the other inputs.

     When called with a single size argument, ‘binornd’ returns a square matrix
     with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     Further information about the binomial distribution can be found at
     <https://en.wikipedia.org/wiki/Binomial_distribution>

     See also: binocdf, binoinv, binopdf, binofit, binolike, binostat, binotest.


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Random arrays from the Binomial distribution.



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bisacdf


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 -- statistics: P = bisacdf (X, BETA, GAMMA)
 -- statistics: P = bisacdf (X, BETA, GAMMA, "upper")

     Birnbaum-Saunders cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the Birnbaum-Saunders distribution with scale parameter BETA and shape
     parameter GAMMA.  The size of P is the common size of X, BETA and GAMMA.  A
     scalar input functions as a constant matrix of the same size as the other
     inputs.

     ‘P = bisacdf (X, BETA, GAMMA, "upper")’ computes the upper tail probability
     of the Birnbaum-Saunders distribution with parameters BETA and GAMMA, at
     the values in X.

     Further information about the Birnbaum-Saunders distribution can be found
     at <https://en.wikipedia.org/wiki/Birnbaum%E2%80%93Saunders_distribution>

     See also: bisainv, bisapdf, bisarnd, bisafit, bisalike, bisastat.


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Birnbaum-Saunders cumulative distribution function (CDF).



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bisainv


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 -- statistics: X = bisainv (P, BETA, GAMMA)

     Inverse of the Birnbaum-Saunders cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     Birnbaum-Saunders distribution with scale parameter BETA and shape
     parameter GAMMA.  The size of X is the common size of P, BETA, and GAMMA.
     A scalar input functions as a constant matrix of the same size as the other
     inputs.

     Further information about the Birnbaum-Saunders distribution can be found
     at <https://en.wikipedia.org/wiki/Birnbaum%E2%80%93Saunders_distribution>

     See also: bisainv, bisapdf, bisarnd, bisafit, bisalike, bisastat.


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Inverse of the Birnbaum-Saunders cumulative distribution function (iCDF).



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bisapdf


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 -- statistics: Y = bisapdf (X, BETA, GAMMA)

     Birnbaum-Saunders probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the Birnbaum-Saunders distribution with scale parameter BETA and shape
     parameter GAMMA.  The size of Y is the common size of X, BETA, and GAMMA.
     A scalar input functions as a constant matrix of the same size as the other
     inputs.

     Further information about the Birnbaum-Saunders distribution can be found
     at <https://en.wikipedia.org/wiki/Birnbaum%E2%80%93Saunders_distribution>

     See also: bisacdf, bisapdf, bisarnd, bisafit, bisalike, bisastat.


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Birnbaum-Saunders probability density function (PDF).



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bisarnd


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 -- statistics: R = bisarnd (BETA, GAMMA)
 -- statistics: R = bisarnd (BETA, GAMMA, ROWS)
 -- statistics: R = bisarnd (BETA, GAMMA, ROWS, COLS, ...)
 -- statistics: R = bisarnd (BETA, GAMMA, [SZ])

     Random arrays from the Birnbaum-Saunders distribution.

     ‘R = bisarnd (BETA, GAMMA)’ returns an array of random numbers chosen from
     the Birnbaum-Saunders distribution with scale parameter BETA and shape
     parameter GAMMA.  The size of R is the common size of BETA and GAMMA.  A
     scalar input functions as a constant matrix of the same size as the other
     inputs.

     When called with a single size argument, ‘bisarnd’ returns a square matrix
     with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     Further information about the Birnbaum-Saunders distribution can be found
     at <https://en.wikipedia.org/wiki/Birnbaum%E2%80%93Saunders_distribution>

     See also: bisacdf, bisainv, bisapdf, bisafit, bisalike, bisastat.


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Random arrays from the Birnbaum-Saunders distribution.



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burrcdf


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 -- statistics: P = burrcdf (X, LAMBDA, C, K)
 -- statistics: P = burrcdf (X, LAMBDA, C, K, "upper")

     Burr type XII cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the Burr type XII distribution with scale parameter LAMBDA, first shape
     parameter C, and second shape parameter K.  The size of P is the common
     size of X, LAMBDA, C, and K.  A scalar input functions as a constant matrix
     of the same size as the other inputs.

     ‘P = burrcdf (X, LAMBDA, C, K, "upper")’ computes the upper tail
     probability of the Burr type XII distribution with parameters LAMBDA, C and
     K, at the values in X.

     Further information about the Burr distribution can be found at
     <https://en.wikipedia.org/wiki/Burr_distribution>

     See also: burrinv, burrpdf, burrrnd, burrfit, burrlike, burrstat.


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Burr type XII cumulative distribution function (CDF).



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burrinv


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 -- statistics: X = burrinv (P, LAMBDA, C, K)

     Inverse of the Burr type XII cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     Burr type XII distribution with scale parameter LAMBDA, first shape
     parameter C, and second shape parameter K.  The size of X is the common
     size of P, LAMBDA, C, and K.  A scalar input functions as a constant matrix
     of the same size as the other inputs.

     Further information about the Burr distribution can be found at
     <https://en.wikipedia.org/wiki/Burr_distribution>

     See also: burrcdf, burrpdf, burrrnd, burrfit, burrlike, burrstat.


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Inverse of the Burr type XII cumulative distribution function (iCDF).



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burrpdf


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 -- statistics: Y = burrpdf (X, LAMBDA, C, K)

     Burr type XII probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the Burr type XII distribution with scale parameter LAMBDA, first shape
     parameter C, and second shape parameter K.  The size of Y is the common
     size of X, LAMBDA, C, and K.  A scalar input functions as a constant matrix
     of the same size as the other inputs.

     Further information about the Burr distribution can be found at
     <https://en.wikipedia.org/wiki/Burr_distribution>

     See also: burrcdf, burrinv, burrrnd, burrfit, burrlike, burrstat.


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Burr type XII probability density function (PDF).



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burrrnd


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 -- statistics: R = burrrnd (LAMBDA, C, K)
 -- statistics: R = burrrnd (LAMBDA, C, K, ROWS)
 -- statistics: R = burrrnd (LAMBDA, C, K, ROWS, COLS, ...)
 -- statistics: R = burrrnd (LAMBDA, C, K, [SZ])

     Random arrays from the Burr type XII distribution.

     ‘R = burrrnd (LAMBDA, C, K)’ returns an array of random numbers chosen from
     the Burr type XII distribution with scale parameter LAMBDA, first shape
     parameter C, and second shape parameter K.  The size of R is the common
     size of LAMBDA, C, and K.  LAMBDA scalar input functions as a constant
     matrix of the same size as the other inputs.

     When called with a single size argument, ‘burrrnd’ returns a square matrix
     with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     Further information about the Burr distribution can be found at
     <https://en.wikipedia.org/wiki/Burr_distribution>

     See also: burrcdf, burrinv, burrpdf, burrfit, burrlike, burrstat.


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Random arrays from the Burr type XII distribution.



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bvncdf


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 -- statistics: P = bvncdf (X, MU, SIGMA)
 -- statistics: P = bvncdf (X, [], SIGMA)

     Bivariate normal cumulative distribution function (CDF).

     ‘P = bvncdf (X, MU, SIGMA)’ will compute the bivariate normal cumulative
     distribution function of X given a mean parameter MU and a scale parameter
     SIGMA.

        • X must be an Nx2 matrix with each variable as a column vector.
        • MU can be either a scalar (common mean) or a two-element row vector
          (each element corresponds to a variable).  If empty, a zero mean is
          assumed.
        • SIGMA can be a scalar (common variance) or a 2x2 covariance matrix,
          which must be positive definite.

     See also: mvncdf.


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Bivariate normal cumulative distribution function (CDF).



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bvtcdf


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 -- statistics: P = bvtcdf (X, RHO, DF)
 -- statistics: P = bvtcdf (X, RHO, DF, TOL)

     Bivariate Student's t cumulative distribution function (CDF).

     ‘P = bvtcdf (X, RHO, DF)’ will compute the bivariate student's t cumulative
     distribution function of X, which must be an Nx2 matrix, given a
     correlation coefficient RHO, which must be a scalar, and DF degrees of
     freedom, which can be a scalar or a vector of positive numbers commensurate
     with X.

     TOL is the tolerance for numerical integration and by default ‘TOL = 1e-8’.

     See also: mvtcdf.


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Bivariate Student's t cumulative distribution function (CDF).



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cauchycdf


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 -- statistics: P = cauchycdf (X, X0, GAMMA)
 -- statistics: P = cauchycdf (X, X0, GAMMA, "upper")

     Cauchy cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the Cauchy distribution with location parameter X0 and scale parameter
     GAMMA.  The size of P is the common size of X, X0, and GAMMA.  A scalar
     input functions as a constant matrix of the same size as the other inputs.

     ‘P = cauchycdf (X, X0, GAMMA, "upper")’ computes the upper tail probability
     of the Cauchy distribution with parameters X0 and GAMMA, at the values in
     X.

     Further information about the Cauchy distribution can be found at
     <https://en.wikipedia.org/wiki/Cauchy_distribution>

     See also: cauchyinv, cauchypdf, cauchyrnd.


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Cauchy cumulative distribution function (CDF).



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cauchyinv


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 -- statistics: X = cauchyinv (P, X0, GAMMA)

     Inverse of the Cauchy cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     Cauchy distribution with location parameter X0 and scale parameter GAMMA.
     The size of X is the common size of P, X0, and GAMMA.  A scalar input
     functions as a constant matrix of the same size as the other inputs.

     Further information about the Cauchy distribution can be found at
     <https://en.wikipedia.org/wiki/Cauchy_distribution>

     See also: cauchycdf, cauchypdf, cauchyrnd.


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Inverse of the Cauchy cumulative distribution function (iCDF).



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cauchypdf


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 -- statistics: Y = cauchypdf (X, X0, GAMMA)

     Cauchy probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the Cauchy distribution with location parameter X0 and scale parameter
     GAMMA.  The size of Y is the common size of X, X0, and GAMMA.  A scalar
     input functions as a constant matrix of the same size as the other inputs.

     Further information about the Cauchy distribution can be found at
     <https://en.wikipedia.org/wiki/Cauchy_distribution>

     See also: cauchycdf, cauchypdf, cauchyrnd.


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Cauchy probability density function (PDF).



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cauchyrnd


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 -- statistics: R = cauchyrnd (X0, GAMMA)
 -- statistics: R = cauchyrnd (X0, GAMMA, ROWS)
 -- statistics: R = cauchyrnd (X0, GAMMA, ROWS, COLS, ...)
 -- statistics: R = cauchyrnd (X0, GAMMA, [SZ])

     Random arrays from the Cauchy distribution.

     ‘R = cauchyrnd (X0, GAMMA)’ returns an array of random numbers chosen from
     the Cauchy distribution with location parameter X0 and scale parameter
     GAMMA.  The size of R is the common size of X0 and GAMMA.  A scalar input
     functions as a constant matrix of the same size as the other inputs.

     When called with a single size argument, ‘cauchyrnd’ returns a square
     matrix with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     Further information about the Cauchy distribution can be found at
     <https://en.wikipedia.org/wiki/Cauchy_distribution>

     See also: cauchycdf, cauchyinv, cauchypdf.


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Random arrays from the Cauchy distribution.



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chi2cdf


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 -- statistics: P = chi2cdf (X, DF)
 -- statistics: P = chi2cdf (X, DF, "upper")

     Chi-squared cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the chi-squared distribution with DF degrees of freedom.  The
     chi-squared density function with DF degrees of freedom is the same as a
     gamma density function with parameters DF/2 and 2.

     The size of P is the common size of X and DF.  A scalar input functions as
     a constant matrix of the same size as the other input.

     ‘P = chi2cdf (X, DF, "upper")’ computes the upper tail probability of the
     chi-squared distribution with DF degrees of freedom, at the values in X.

     Further information about the chi-squared distribution can be found at
     <https://en.wikipedia.org/wiki/Chi-squared_distribution>

     See also: chi2inv, chi2pdf, chi2rnd, chi2stat.


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Chi-squared cumulative distribution function (CDF).



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chi2inv


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 -- statistics: X = chi2inv (P, DF)

     Inverse of the chi-squared cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     chi-squared distribution with DF degrees of freedom.  The size of X is the
     common size of P and DF.  A scalar input functions as a constant matrix of
     the same size as the other inputs.

     Further information about the chi-squared distribution can be found at
     <https://en.wikipedia.org/wiki/Chi-squared_distribution>

     See also: chi2cdf, chi2pdf, chi2rnd, chi2stat.


# name: <cell-element>
# type: sq_string
# elements: 1
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Inverse of the chi-squared cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
chi2pdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 564
 -- statistics: Y = chi2pdf (X, DF)

     Chi-squared probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the chi-squared distribution with DF degrees of freedom.  The size of Y is
     the common size of X and DF.  A scalar input functions as a constant matrix
     of the same size as the other inputs.

     Further information about the chi-squared distribution can be found at
     <https://en.wikipedia.org/wiki/Chi-squared_distribution>

     See also: chi2cdf, chi2pdf, chi2rnd, chi2stat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 47
Chi-squared probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
chi2rnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 968
 -- statistics: R = chi2rnd (DF)
 -- statistics: R = chi2rnd (DF, ROWS)
 -- statistics: R = chi2rnd (DF, ROWS, COLS, ...)
 -- statistics: R = chi2rnd (DF, [SZ])

     Random arrays from the chi-squared distribution.

     ‘R = chi2rnd (DF)’ returns an array of random numbers chosen from the
     chi-squared distribution with DF degrees of freedom.  The size of R is the
     size of DF.

     When called with a single size argument, ‘chi2rnd’ returns a square matrix
     with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     Further information about the chi-squared distribution can be found at
     <https://en.wikipedia.org/wiki/Chi-squared_distribution>

     See also: chi2cdf, chi2inv, chi2pdf, chi2stat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 48
Random arrays from the chi-squared distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 9
copulacdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 2243
 -- statistics: P = copulacdf (FAMILY, X, THETA)
 -- statistics: P = copulacdf ('t', X, THETA, DF)

     Copula family cumulative distribution functions (CDF).

     Arguments
     ---------

        • FAMILY is the copula family name.  Currently, FAMILY can be
          ‘'Gaussian'’ for the Gaussian family, ‘'t'’ for the Student's t
          family, ‘'Clayton'’ for the Clayton family, ‘'Gumbel'’ for the
          Gumbel-Hougaard family, ‘'Frank'’ for the Frank family, ‘'AMH'’ for
          the Ali-Mikhail-Haq family, or ‘'FGM'’ for the
          Farlie-Gumbel-Morgenstern family.

        • X is the support where each row corresponds to an observation.

        • THETA is the parameter of the copula.  For the Gaussian and Student's
          t copula, THETA must be a correlation matrix.  For bivariate copulas
          THETA can also be a correlation coefficient.  For the Clayton family,
          the Gumbel-Hougaard family, the Frank family, and the Ali-Mikhail-Haq
          family, THETA must be a vector with the same number of elements as
          observations in X or be scalar.  For the Farlie-Gumbel-Morgenstern
          family, THETA must be a matrix of coefficients for the
          Farlie-Gumbel-Morgenstern polynomial where each row corresponds to one
          set of coefficients for an observation in X.  A single row is
          expanded.  The coefficients are in binary order.

        • DF is the degrees of freedom for the Student's t family.  DF must be a
          vector with the same number of elements as observations in X or be
          scalar.

     Return values
     -------------

        • P is the cumulative distribution of the copula at each row of X and
          corresponding parameter THETA.

     Examples
     --------

          x = [0.2:0.2:0.6; 0.2:0.2:0.6];
          theta = [1; 2];
          p = copulacdf ("Clayton", x, theta)

          x = [0.2:0.2:0.6; 0.2:0.1:0.4];
          theta = [0.2, 0.1, 0.1, 0.05];
          p = copulacdf ("FGM", x, theta)

     References
     ----------

       1. Roger B. Nelsen.  ‘An Introduction to Copulas’.  Springer, New York,
          second edition, 2006.

     See also: copulapdf, copularnd.


# name: <cell-element>
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Copula family cumulative distribution functions (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 9
copulapdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1559
 -- statistics: Y = copulapdf (FAMILY, X, THETA)

     Copula family probability density functions (PDF).

     Arguments
     ---------

        • FAMILY is the copula family name.  Currently, FAMILY can be
          ‘'Clayton'’ for the Clayton family, ‘'Gumbel'’ for the Gumbel-Hougaard
          family, ‘'Frank'’ for the Frank family, or ‘'AMH'’ for the
          Ali-Mikhail-Haq family.

        • X is the support where each row corresponds to an observation.

        • THETA is the parameter of the copula.  The elements of THETA must be
          greater than or equal to ‘-1’ for the Clayton family, greater than or
          equal to ‘1’ for the Gumbel-Hougaard family, arbitrary for the Frank
          family, and greater than or equal to ‘-1’ and lower than ‘1’ for the
          Ali-Mikhail-Haq family.  Moreover, THETA must be non-negative for
          dimensions greater than ‘2’.  THETA must be a column vector with the
          same number of rows as X or be scalar.

     Return values
     -------------

        • Y is the probability density of the copula at each row of X and
          corresponding parameter THETA.

     Examples
     --------

          x = [0.2:0.2:0.6; 0.2:0.2:0.6];
          theta = [1; 2];
          y = copulapdf ("Clayton", x, theta)

          y = copulapdf ("Gumbel", x, 2)

     References
     ----------

       1. Roger B. Nelsen.  ‘An Introduction to Copulas’.  Springer, New York,
          second edition, 2006.

     See also: copulacdf, copularnd.


# name: <cell-element>
# type: sq_string
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# length: 50
Copula family probability density functions (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 9
copularnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1920
 -- Function File: R = copularnd (FAMILY, THETA, N)
 -- Function File: R = copularnd (FAMILY, THETA, N, D)
 -- Function File: R = copularnd ('t', THETA, DF, N)

     Random arrays from the copula family distributions.

     Arguments
     ---------

        • FAMILY is the copula family name.  Currently, FAMILY can be
          ‘'Gaussian'’ for the Gaussian family, ‘'t'’ for the Student's t
          family, or ‘'Clayton'’ for the Clayton family.

        • THETA is the parameter of the copula.  For the Gaussian and Student's
          t copula, THETA must be a correlation matrix.  For bivariate copulas
          THETA can also be a correlation coefficient.  For the Clayton family,
          THETA must be a vector with the same number of elements as samples to
          be generated or be scalar.

        • DF is the degrees of freedom for the Student's t family.  DF must be a
          vector with the same number of elements as samples to be generated or
          be scalar.

        • N is the number of rows of the matrix to be generated.  N must be a
          non-negative integer and corresponds to the number of samples to be
          generated.

        • D is the number of columns of the matrix to be generated.  D must be a
          positive integer and corresponds to the dimension of the copula.

     Return values
     -------------

        • R is a matrix of random samples from the copula with N samples of
          distribution dimension D.

     Examples
     --------

          theta = 0.5;
          r = copularnd ("Gaussian", theta);

          theta = 0.5;
          df = 2;
          r = copularnd ("t", theta, df);

          theta = 0.5;
          n = 2;
          r = copularnd ("Clayton", theta, n);

     References
     ----------

       1. Roger B. Nelsen.  ‘An Introduction to Copulas’.  Springer, New York,
          second edition, 2006.


# name: <cell-element>
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# length: 51
Random arrays from the copula family distributions.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
evcdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 2031
 -- statistics: P = evcdf (X)
 -- statistics: P = evcdf (X, MU)
 -- statistics: P = evcdf (X, MU, SIGMA)
 -- statistics: P = evcdf (..., "upper")
 -- statistics: [P, PLO, PUP] = evcdf (X, MU, SIGMA, PCOV)
 -- statistics: [P, PLO, PUP] = evcdf (X, MU, SIGMA, PCOV, ALPHA)
 -- statistics: [P, PLO, PUP] = evcdf (..., "upper")

     Extreme value cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the extreme value distribution (also known as the Gumbel or the type I
     generalized extreme value distribution) at the values in X with location
     parameter MU and scale parameter SIGMA.  The size of P is the common size
     of X, MU and SIGMA.  A scalar input functions as a constant matrix of the
     same size as the other inputs.

     Default values are MU = 0 and SIGMA = 1.

     When called with three output arguments, i.e.  [P, PLO, PUP], ‘evcdf’
     computes the confidence bounds for P when the input parameters MU and SIGMA
     are estimates.  In such case, PCOV, a 2x2 matrix containing the covariance
     matrix of the estimated parameters, is necessary.  Optionally, ALPHA, which
     has a default value of 0.05, specifies the 100 * (1 - ALPHA) percent
     confidence bounds.  PLO and PUP are arrays of the same size as P containing
     the lower and upper confidence bounds.

     ‘[...] = evcdf (..., "upper")’ computes the upper tail probability of the
     extreme value distribution with parameters X0 and GAMMA, at the values in
     X.

     The Gumbel distribution is used to model the distribution of the maximum
     (or the minimum) of a number of samples of various distributions.  This
     version is suitable for modeling minima.  For modeling maxima, use the
     alternative Gumbel CDF, ‘gumbelcdf’.

     Further information about the Gumbel distribution can be found at
     <https://en.wikipedia.org/wiki/Gumbel_distribution>

     See also: evinv, evpdf, evrnd, evfit, evlike, evstat, gumbelcdf.


# name: <cell-element>
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# elements: 1
# length: 53
Extreme value cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
evinv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1759
 -- statistics: X = evinv (P)
 -- statistics: X = evinv (P, MU)
 -- statistics: X = evinv (P, MU, SIGMA)
 -- statistics: [X, XLO, XUP] = evinv (P, MU, SIGMA, PCOV)
 -- statistics: [X, XLO, XUP] = evinv (P, MU, SIGMA, PCOV, ALPHA)

     Inverse of the extreme value cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     extreme value distribution (also known as the Gumbel or the type I
     generalized extreme value distribution) with location parameter MU and
     scale parameter SIGMA.  The size of X is the common size of P, MU and
     SIGMA.  A scalar input functions as a constant matrix of the same size as
     the other inputs.

     Default values are MU = 0 and SIGMA = 1.

     When called with three output arguments, i.e.  [X, XLO, XUP], ‘evinv’
     computes the confidence bounds for X when the input parameters MU and SIGMA
     are estimates.  In such case, PCOV, a 2x2 matrix containing the covariance
     matrix of the estimated parameters, is necessary.  Optionally, ALPHA, which
     has a default value of 0.05, specifies the 100 * (1 - ALPHA) percent
     confidence bounds.  XLO and XUP are arrays of the same size as X containing
     the lower and upper confidence bounds.

     The Gumbel distribution is used to model the distribution of the maximum
     (or the minimum) of a number of samples of various distributions.  This
     version is suitable for modeling minima.  For modeling maxima, use the
     alternative Gumbel iCDF, ‘gumbelinv’.

     Further information about the Gumbel distribution can be found at
     <https://en.wikipedia.org/wiki/Gumbel_distribution>

     See also: evcdf, evpdf, evrnd, evfit, evlike, evstat, gumbelinv.


# name: <cell-element>
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# elements: 1
# length: 69
Inverse of the extreme value cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
evpdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1094
 -- statistics: Y = evpdf (X)
 -- statistics: Y = evpdf (X, MU)
 -- statistics: Y = evpdf (X, MU, SIGMA)

     Extreme value probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the extreme value distribution (also known as the Gumbel or the type I
     generalized extreme value distribution) with location parameter MU and
     scale parameter SIGMA.  The size of Y is the common size of X, MU and
     SIGMA.  A scalar input functions as a constant matrix of the same size as
     the other inputs.

     Default values are MU = 0 and SIGMA = 1.

     The Gumbel distribution is used to model the distribution of the maximum
     (or the minimum) of a number of samples of various distributions.  This
     version is suitable for modeling minima.  For modeling maxima, use the
     alternative Gumbel iCDF, ‘gumbelinv’.

     Further information about the Gumbel distribution can be found at
     <https://en.wikipedia.org/wiki/Gumbel_distribution>

     See also: evcdf, evinv, evrnd, evfit, evlike, evstat, gumbelpdf.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 49
Extreme value probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
evrnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1493
 -- statistics: R = evrnd (MU, SIGMA)
 -- statistics: R = evrnd (MU, SIGMA, ROWS)
 -- statistics: R = evrnd (MU, SIGMA, ROWS, COLS, ...)
 -- statistics: R = evrnd (MU, SIGMA, [SZ])

     Random arrays from the extreme value distribution.

     ‘R = evrnd (MU, SIGMA)’ returns an array of random numbers chosen from the
     extreme value distribution (also known as the Gumbel or the type I
     generalized extreme value distribution) with location parameter MU and
     scale parameter SIGMA.  The size of R is the common size of MU and SIGMA.
     A scalar input functions as a constant matrix of the same size as the other
     inputs.

     When called with a single size argument, ‘evrnd’ returns a square matrix
     with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     The Gumbel distribution is used to model the distribution of the maximum
     (or the minimum) of a number of samples of various distributions.  This
     version is suitable for modeling minima.  For modeling maxima, use the
     alternative Gumbel iCDF, ‘gumbelinv’.

     Further information about the Gumbel distribution can be found at
     <https://en.wikipedia.org/wiki/Gumbel_distribution>

     See also: evcdf, evinv, evpdf, evfit, evlike, evstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 50
Random arrays from the extreme value distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
expcdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1806
 -- statistics: P = expcdf (X)
 -- statistics: P = expcdf (X, MU)
 -- statistics: P = expcdf (..., "upper")
 -- statistics: [P, PLO, PUP] = expcdf (X, MU, PCOV)
 -- statistics: [P, PLO, PUP] = expcdf (X, MU, PCOV, ALPHA)
 -- statistics: [P, PLO, PUP] = expcdf (..., "upper")

     Exponential cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the exponential distribution with mean parameter MU.  The size of P is
     the common size of X and MU.  A scalar input functions as a constant matrix
     of the same size as the other inputs.

     Default value is MU = 1.

     A common alternative parameterization of the exponential distribution is to
     use the parameter λ defined as the mean number of events in an interval as
     opposed to the parameter μ, which is the mean wait time for an event to
     occur.  λ and μ are reciprocals, i.e.  μ = 1 / λ.

     When called with three output arguments, i.e.  [P, PLO, PUP], ‘expcdf’
     computes the confidence bounds for P when the input parameter MU is an
     estimate.  In such case, PCOV, a scalar value with the variance of the
     estimated parameter MU, is necessary.  Optionally, ALPHA, which has a
     default value of 0.05, specifies the 100 * (1 - ALPHA) percent confidence
     bounds.  PLO and PUP are arrays of the same size as P containing the lower
     and upper confidence bounds.

     ‘[...] = expcdf (..., "upper")’ computes the upper tail probability of the
     exponential distribution with parameter MU, at the values in X.

     Further information about the exponential distribution can be found at
     <https://en.wikipedia.org/wiki/Exponential_distribution>

     See also: expinv, exppdf, exprnd, expfit, explike, expstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 51
Exponential cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
expinv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1557
 -- statistics: X = expinv (P)
 -- statistics: X = expinv (P, MU)
 -- statistics: [X, XLO, XUP] = expinv (P, MU, PCOV)
 -- statistics: [X, XLO, XUP] = expinv (P, MU, PCOV, ALPHA)

     Inverse of the exponential cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     exponential distribution with mean MU.  The size of X is the common size of
     P and MU.  A scalar input functions as a constant matrix of the same size
     as the other inputs.

     Default value is MU = 1.

     A common alternative parameterization of the exponential distribution is to
     use the parameter λ defined as the mean number of events in an interval as
     opposed to the parameter μ, which is the mean wait time for an event to
     occur.  λ and μ are reciprocals, i.e.  μ = 1 / λ.

     When called with three output arguments, i.e.  [X, XLO, XUP], ‘expinv’
     computes the confidence bounds for X when the input parameter MU is an
     estimate.  In such case, PCOV, a scalar value with the variance of the
     estimated parameter MU, is necessary.  Optionally, ALPHA, which has a
     default value of 0.05, specifies the 100 * (1 - ALPHA) percent confidence
     bounds.  XLO and XUP are arrays of the same size as X containing the lower
     and upper confidence bounds.

     Further information about the exponential distribution can be found at
     <https://en.wikipedia.org/wiki/Exponential_distribution>

     See also: expcdf, exppdf, exprnd, expfit, explike, expstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 67
Inverse of the exponential cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
exppdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 935
 -- statistics: Y = exppdf (X)
 -- statistics: Y = exppdf (X, MU)

     Exponential probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the exponential distribution with mean parameter MU.  The size of Y is the
     common size of X and MU.  A scalar input functions as a constant matrix of
     the same size as the other inputs.

     Default value for MU = 1.

     A common alternative parameterization of the exponential distribution is to
     use the parameter λ defined as the mean number of events in an interval as
     opposed to the parameter μ, which is the mean wait time for an event to
     occur.  λ and μ are reciprocals, i.e.  μ = 1 / λ.

     Further information about the exponential distribution can be found at
     <https://en.wikipedia.org/wiki/Exponential_distribution>

     See also: expcdf, expinv, exprnd, expfit, explike, expstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 47
Exponential probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
exprnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1271
 -- statistics: R = exprnd (MU)
 -- statistics: R = exprnd (MU, ROWS)
 -- statistics: R = exprnd (MU, ROWS, COLS, ...)
 -- statistics: R = exprnd (MU, [SZ])

     Random arrays from the exponential distribution.

     ‘R = exprnd (MU)’ returns an array of random numbers chosen from the
     exponential distribution with mean parameter MU.  The size of R is the size
     of MU.

     When called with a single size argument, ‘exprnd’ returns a square matrix
     with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     A common alternative parameterization of the exponential distribution is to
     use the parameter λ defined as the mean number of events in an interval as
     opposed to the parameter μ, which is the mean wait time for an event to
     occur.  λ and μ are reciprocals, i.e.  μ = 1 / λ.

     Further information about the exponential distribution can be found at
     <https://en.wikipedia.org/wiki/Exponential_distribution>

     See also: expcdf, expinv, exppdf, expfit, explike, expstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 48
Random arrays from the exponential distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 4
fcdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 751
 -- statistics: P = fcdf (X, DF1, DF2)
 -- statistics: P = fcdf (X, DF1, DF2, "upper")

     F-cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the F-distribution with DF1 and DF2 degrees of freedom.  The size of P
     is the common size of X, DF1, and DF2.  A scalar input functions as a
     constant matrix of the same size as the other inputs.

     ‘P = fcdf (X, DF1, DF2, "upper")’ computes the upper tail probability of
     the F-distribution with DF1 and DF2 degrees of freedom, at the values in X.

     Further information about the F-distribution can be found at
     <https://en.wikipedia.org/wiki/F-distribution>

     See also: finv, fpdf, frnd, fstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 41
F-cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 4
finv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 550
 -- statistics: X = finv (P, DF1, DF2)

     Inverse of the F-cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     F-distribution with DF1 and DF2 degrees of freedom.  The size of X is the
     common size of P, DF1, and DF2.  A scalar input functions as a constant
     matrix of the same size as the other inputs.

     Further information about the F-distribution can be found at
     <https://en.wikipedia.org/wiki/F-distribution>

     See also: fcdf, fpdf, frnd, fstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 57
Inverse of the F-cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 4
fpdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 531
 -- statistics: Y = fpdf (X, DF1, DF2)

     F-probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the F-distribution with DF1 and DF2 degrees of freedom.  The size of Y is
     the common size of X, DF1, and DF2.  A scalar input functions as a constant
     matrix of the same size as the other inputs.

     Further information about the F-distribution can be found at
     <https://en.wikipedia.org/wiki/F-distribution>

     See also: fcdf, finv, frnd, fstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 37
F-probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 4
frnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1043
 -- statistics: R = frnd (DF1, DF2)
 -- statistics: R = frnd (DF1, DF2, ROWS)
 -- statistics: R = frnd (DF1, DF2, ROWS, COLS, ...)
 -- statistics: R = frnd (DF1, DF2, [SZ])

     Random arrays from the F-distribution.

     ‘R = frnd (DF1, DF2)’ returns an array of random numbers chosen from the
     F-distribution with DF1 and DF2 degrees of freedom.  The size of R is the
     common size of DF1 and DF2.  A scalar input functions as a constant matrix
     of the same size as the other inputs.

     When called with a single size argument, ‘frnd’ returns a square matrix
     with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     Further information about the F-distribution can be found at
     <https://en.wikipedia.org/wiki/F-distribution>

     See also: fcdf, finv, fpdf, fstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 38
Random arrays from the F-distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
gamcdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1991
 -- statistics: P = gamcdf (X, A)
 -- statistics: P = gamcdf (X, A, B)
 -- statistics: P = gamcdf (..., "upper")
 -- statistics: [P, PLO, PUP] = gamcdf (X, A, B, PCOV)
 -- statistics: [P, PLO, PUP] = gamcdf (X, A, B, PCOV, ALPHA)
 -- statistics: [P, PLO, PUP] = gamcdf (..., "upper")

     Gamma cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the Gamma distribution with shape parameter A and scale parameter B.
     When called with only one parameter, then B defaults to 1.  The size of P
     is the common size of X, A, and B.  A scalar input functions as a constant
     matrix of the same size as the other inputs.

     When called with three output arguments, i.e.  [P, PLO, PUP], ‘gamcdf’
     computes the confidence bounds for P when the input parameters A and B are
     estimates.  In such case, PCOV, a 2x2 matrix containing the covariance
     matrix of the estimated parameters, is necessary.  Optionally, ALPHA, which
     has a default value of 0.05, specifies the 100 * (1 - ALPHA) percent
     confidence bounds.  PLO and PUP are arrays of the same size as P containing
     the lower and upper confidence bounds.

     ‘[...] = gamcdf (..., "upper")’ computes the upper tail probability of the
     Gamma distribution with parameters A and B, at the values in X.

     OCTAVE/MATLAB use the alternative parameterization given by the pair α, β,
     i.e.  shape A and scale B.  In Wikipedia, the two common parameterizations
     use the pairs k, θ, as shape and scale, and α, β, as shape and rate,
     respectively.  The parameter names A and B used here (for MATLAB
     compatibility) correspond to the parameter notation k, θ instead of the α,
     β as reported in Wikipedia.

     Further information about the Gamma distribution can be found at
     <https://en.wikipedia.org/wiki/Gamma_distribution>

     See also: gaminv, gampdf, gamrnd, gamfit, gamlike, gamstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 45
Gamma cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
gaminv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1020
 -- statistics: X = gaminv (P, A, B)

     Inverse of the Gamma cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     Gamma distribution with shape parameter A and scale parameter B.  The size
     of X is the common size of P, A, and B.  A scalar input functions as a
     constant matrix of the same size as the other inputs.

     OCTAVE/MATLAB use the alternative parameterization given by the pair α, β,
     i.e.  shape A and scale B.  In Wikipedia, the two common parameterizations
     use the pairs k, θ, as shape and scale, and α, β, as shape and rate,
     respectively.  The parameter names A and B used here (for MATLAB
     compatibility) correspond to the parameter notation k, θ instead of the α,
     β as reported in Wikipedia.

     Further information about the Gamma distribution can be found at
     <https://en.wikipedia.org/wiki/Gamma_distribution>

     See also: gamcdf, gampdf, gamrnd, gamfit, gamlike, gamstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 61
Inverse of the Gamma cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
gampdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1000
 -- statistics: Y = gampdf (X, A, B)

     Gamma probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the Gamma distribution with shape parameter A and scale parameter B.  The
     size of Y is the common size of X, A and B.  A scalar input functions as a
     constant matrix of the same size as the other inputs.

     OCTAVE/MATLAB use the alternative parameterization given by the pair α, β,
     i.e.  shape A and scale B.  In Wikipedia, the two common parameterizations
     use the pairs k, θ, as shape and scale, and α, β, as shape and rate,
     respectively.  The parameter names A and B used here (for MATLAB
     compatibility) correspond to the parameter notation k, θ instead of the α,
     β as reported in Wikipedia.

     Further information about the Gamma distribution can be found at
     <https://en.wikipedia.org/wiki/Gamma_distribution>

     See also: gamcdf, gaminv, gamrnd, gamfit, gamlike, gamstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 41
Gamma probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
gamrnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1507
 -- statistics: R = gamrnd (A, B)
 -- statistics: R = gamrnd (A, B, ROWS)
 -- statistics: R = gamrnd (A, B, ROWS, COLS, ...)
 -- statistics: R = gamrnd (A, B, [SZ])

     Random arrays from the Gamma distribution.

     ‘R = gamrnd (A, B)’ returns an array of random numbers chosen from the
     Gamma distribution with shape parameter A and scale parameter B.  The size
     of R is the common size of A and B.  A scalar input functions as a constant
     matrix of the same size as the other inputs.

     When called with a single size argument, ‘gamrnd’ returns a square matrix
     with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     OCTAVE/MATLAB use the alternative parameterization given by the pair α, β,
     i.e.  shape A and scale B.  In Wikipedia, the two common parameterizations
     use the pairs k, θ, as shape and scale, and α, β, as shape and rate,
     respectively.  The parameter names A and B used here (for MATLAB
     compatibility) correspond to the parameter notation k, θ instead of the α,
     β as reported in Wikipedia.

     Further information about the Gamma distribution can be found at
     <https://en.wikipedia.org/wiki/Gamma_distribution>

     See also: gamcdf, gaminv, gampdf, gamfit, gamlike, gamstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 42
Random arrays from the Gamma distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
geocdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 913
 -- statistics: P = geocdf (X, PS)
 -- statistics: P = geocdf (X, PS, "upper")

     Geometric cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the geometric distribution with probability of success parameter PS.
     The size of P is the common size of X and PS.  A scalar input functions as
     a constant matrix of the same size as the other inputs.

     ‘P = geocdf (X, PS, "upper")’ computes the upper tail probability of the
     geometric distribution with parameter PS, at the values in X.

     The geometric distribution models the number of failures (X) of a Bernoulli
     trial with probability PS before the first success.

     Further information about the geometric distribution can be found at
     <https://en.wikipedia.org/wiki/Geometric_distribution>

     See also: geoinv, geopdf, geornd, geofit, geostat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 49
Geometric cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
geoinv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 731
 -- statistics: X = geoinv (P, PS)

     Inverse of the geometric cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     geometric distribution with probability of success parameter PS.  The size
     of X is the common size of P and PS.  A scalar input functions as a
     constant matrix of the same size as the other inputs.

     The geometric distribution models the number of failures (P) of a Bernoulli
     trial with probability PS before the first success.

     Further information about the geometric distribution can be found at
     <https://en.wikipedia.org/wiki/Geometric_distribution>

     See also: geocdf, geopdf, geornd, geofit, geostat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 65
Inverse of the geometric cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
geopdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 712
 -- statistics: Y = geopdf (X, PS)

     Geometric probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the geometric distribution with probability of success parameter PS.  The
     size of Y is the common size of X and PS.  A scalar input functions as a
     constant matrix of the same size as the other inputs.

     The geometric distribution models the number of failures (X) of a Bernoulli
     trial with probability PS before the first success.

     Further information about the geometric distribution can be found at
     <https://en.wikipedia.org/wiki/Geometric_distribution>

     See also: geocdf, geoinv, geornd, geofit, geostat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 45
Geometric probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
geornd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1118
 -- statistics: R = geornd (PS)
 -- statistics: R = geornd (PS, ROWS)
 -- statistics: R = geornd (PS, ROWS, COLS, ...)
 -- statistics: R = geornd (PS, [SZ])

     Random arrays from the geometric distribution.

     ‘R = geornd (PS)’ returns an array of random numbers chosen from the
     Birnbaum-Saunders distribution with probability of success parameter PS.
     The size of R is the size of PS.

     When called with a single size argument, ‘geornd’ returns a square matrix
     with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     The geometric distribution models the number of failures (X) of a Bernoulli
     trial with probability PS before the first success.

     Further information about the geometric distribution can be found at
     <https://en.wikipedia.org/wiki/Geometric_distribution>

     See also: geocdf, geoinv, geopdf, geofit, geostat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 46
Random arrays from the geometric distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
gevcdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1664
 -- statistics: P = gevcdf (X, K, SIGMA, MU)
 -- statistics: P = gevcdf (X, K, SIGMA, MU, "upper")

     Generalized extreme value (GEV) cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the GEV distribution with shape parameter K, scale parameter SIGMA, and
     location parameter MU.  The size of P is the common size of X, K, SIGMA,
     and MU.  A scalar input functions as a constant matrix of the same size as
     the other inputs.

     ‘[...] = gevcdf (X, K, SIGMA, MU, "upper")’ computes the upper tail
     probability of the GEV distribution with parameters K, SIGMA, and MU, at
     the values in X.

     When K < 0, the GEV is the type III extreme value distribution.  When K >
     0, the GEV distribution is the type II, or Frechet, extreme value
     distribution.  If W has a Weibull distribution as computed by the ‘wblcdf’
     function, then -W has a type III extreme value distribution and 1/W has a
     type II extreme value distribution.  In the limit as K approaches 0, the
     GEV is the mirror image of the type I extreme value distribution as
     computed by the ‘evcdf’ function.

     The mean of the GEV distribution is not finite when K >= 1, and the
     variance is not finite when K >= 1/2.  The GEV distribution has positive
     density only for values of X such that K * (X - MU) / SIGMA > -1.

     Further information about the generalized extreme value distribution can be
     found at
     <https://en.wikipedia.org/wiki/Generalized_extreme_value_distribution>

     See also: gevinv, gevpdf, gevrnd, gevfit, gevlike, gevstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 71
Generalized extreme value (GEV) cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
gevinv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1448
 -- statistics: X = gevinv (P, K, SIGMA, MU)

     Inverse of the generalized extreme value (GEV) cumulative distribution
     function (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     GEV distribution with shape parameter K, scale parameter SIGMA, and
     location parameter MU.  The size of P is the common size of X, K, SIGMA,
     and MU.  A scalar input functions as a constant matrix of the same size as
     the other inputs.

     When K < 0, the GEV is the type III extreme value distribution.  When K >
     0, the GEV distribution is the type II, or Frechet, extreme value
     distribution.  If W has a Weibull distribution as computed by the ‘wblcdf’
     function, then -W has a type III extreme value distribution and 1/W has a
     type II extreme value distribution.  In the limit as K approaches 0, the
     GEV is the mirror image of the type I extreme value distribution as
     computed by the ‘evcdf’ function.

     The mean of the GEV distribution is not finite when K >= 1, and the
     variance is not finite when K >= 1/2.  The GEV distribution has positive
     density only for values of X such that K * (X - MU) / SIGMA > -1.

     Further information about the generalized extreme value distribution can be
     found at
     <https://en.wikipedia.org/wiki/Generalized_extreme_value_distribution>

     See also: gevcdf, gevpdf, gevrnd, gevfit, gevlike, gevstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Inverse of the generalized extreme value (GEV) cumulative distribution functi...



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
gevpdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1424
 -- statistics: Y = gevpdf (X, K, SIGMA, MU)

     Generalized extreme value (GEV) probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the GEV distribution with shape parameter K, scale parameter SIGMA, and
     location parameter MU.  The size of Y is the common size of X, K, SIGMA,
     and MU.  A scalar input functions as a constant matrix of the same size as
     the other inputs.

     When K < 0, the GEV is the type III extreme value distribution.  When K >
     0, the GEV distribution is the type II, or Frechet, extreme value
     distribution.  If W has a Weibull distribution as computed by the ‘wblcdf’
     function, then -W has a type III extreme value distribution and 1/W has a
     type II extreme value distribution.  In the limit as K approaches 0, the
     GEV is the mirror image of the type I extreme value distribution as
     computed by the ‘evcdf’ function.

     The mean of the GEV distribution is not finite when K >= 1, and the
     variance is not finite when K >= 1/2.  The GEV distribution has positive
     density only for values of X such that K * (X - MU) / SIGMA > -1.

     Further information about the generalized extreme value distribution can be
     found at
     <https://en.wikipedia.org/wiki/Generalized_extreme_value_distribution>

     See also: gevcdf, gevinv, gevrnd, gevfit, gevlike, gevstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 67
Generalized extreme value (GEV) probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
gevrnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1962
 -- statistics: R = gevrnd (K, SIGMA, MU)
 -- statistics: R = gevrnd (K, SIGMA, MU, ROWS)
 -- statistics: R = gevrnd (K, SIGMA, MU, ROWS, COLS, ...)
 -- statistics: R = gevrnd (K, SIGMA, MU, [SZ])

     Random arrays from the generalized extreme value (GEV) distribution.

     ‘R = gevrnd (K, SIGMA, MU’ returns an array of random numbers chosen from
     the GEV distribution with shape parameter K, scale parameter SIGMA, and
     location parameter MU.  The size of R is the common size of K, SIGMA, and
     MU.  A scalar input functions as a constant matrix of the same size as the
     other inputs.

     When called with a single size argument, ‘gevrnd’ returns a square matrix
     with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     When K < 0, the GEV is the type III extreme value distribution.  When K >
     0, the GEV distribution is the type II, or Frechet, extreme value
     distribution.  If W has a Weibull distribution as computed by the ‘wblcdf’
     function, then -W has a type III extreme value distribution and 1/W has a
     type II extreme value distribution.  In the limit as K approaches 0, the
     GEV is the mirror image of the type I extreme value distribution as
     computed by the ‘evcdf’ function.

     The mean of the GEV distribution is not finite when K >= 1, and the
     variance is not finite when K >= 1/2.  The GEV distribution has positive
     density only for values of X such that K * (X - MU) / SIGMA > -1.

     Further information about the generalized extreme value distribution can be
     found at
     <https://en.wikipedia.org/wiki/Generalized_extreme_value_distribution>

     See also: gevcdf, gevinv, gevpdf, gevfit, gevlike, gevstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 68
Random arrays from the generalized extreme value (GEV) distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
gpcdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1410
 -- statistics: P = gpcdf (X, K, SIGMA, THETA)
 -- statistics: P = gpcdf (X, K, SIGMA, THETA, "upper")

     Generalized Pareto cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the generalized Pareto distribution with shape parameter K, scale
     parameter SIGMA, and location parameter THETA.  The size of P is the common
     size of X, K, SIGMA, and THETA.  A scalar input functions as a constant
     matrix of the same size as the other inputs.

     ‘[...] = gpcdf(X, K, SIGMA, THETA, "upper")’ computes the upper tail
     probability of the generalized Pareto distribution with parameters K,
     SIGMA, and THETA, at the values in X.

     When K = 0 and THETA = 0, the Generalized Pareto is equivalent to the
     exponential distribution.  When K > 0 and ‘THETA = K / K’ the Generalized
     Pareto is equivalent τπ the Pareto distribution.  The mean of the
     Generalized Pareto is not finite when K >= 1 and the variance is not finite
     when K >= 1/2.  When K >= 0, the Generalized Pareto has positive density
     for X > THETA, or, when THETA < 0, for 0 <= (X - THETA) / SIGMA <= -1 / K.

     Further information about the generalized Pareto distribution can be found
     at <https://en.wikipedia.org/wiki/Generalized_Pareto_distribution>

     See also: gpinv, gppdf, gprnd, gpfit, gplike, gpstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 58
Generalized Pareto cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
gpinv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1166
 -- statistics: X = gpinv (P, K, SIGMA, THETA)

     Inverse of the generalized Pareto cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     generalized Pareto distribution with shape parameter K, scale parameter
     SIGMA, and location parameter THETA.  The size of X is the common size of
     P, K, SIGMA, and THETA.  A scalar input functions as a constant matrix of
     the same size as the other inputs.

     When K = 0 and THETA = 0, the Generalized Pareto is equivalent to the
     exponential distribution.  When K > 0 and ‘THETA = K / K’ the Generalized
     Pareto is equivalent to the Pareto distribution.  The mean of the
     Generalized Pareto is not finite when K >= 1 and the variance is not finite
     when K >= 1/2.  When K >= 0, the Generalized Pareto has positive density
     for X > THETA, or, when THETA < 0, for 0 <= (X - THETA) / SIGMA <= -1 / K.

     Further information about the generalized Pareto distribution can be found
     at <https://en.wikipedia.org/wiki/Generalized_Pareto_distribution>

     See also: gpcdf, gppdf, gprnd, gpfit, gplike, gpstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 74
Inverse of the generalized Pareto cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
gppdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1147
 -- statistics: Y = gppdf (X, K, SIGMA, THETA)

     Generalized Pareto probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the generalized Pareto distribution with shape parameter K, scale parameter
     SIGMA, and location parameter THETA.  The size of Y is the common size of
     P, K, SIGMA, and THETA.  A scalar input functions as a constant matrix of
     the same size as the other inputs.

     When K = 0 and THETA = 0, the Generalized Pareto is equivalent to the
     exponential distribution.  When K > 0 and ‘THETA = K / K’ the Generalized
     Pareto is equivalent to the Pareto distribution.  The mean of the
     Generalized Pareto is not finite when K >= 1 and the variance is not finite
     when K >= 1/2.  When K >= 0, the Generalized Pareto has positive density
     for X > THETA, or, when THETA < 0, for 0 <= (X - THETA) / SIGMA <= -1 / K.

     Further information about the generalized Pareto distribution can be found
     at <https://en.wikipedia.org/wiki/Generalized_Pareto_distribution>

     See also: gpcdf, gpinv, gprnd, gpfit, gplike, gpstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 54
Generalized Pareto probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
gprnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1693
 -- statistics: R = gprnd (K, SIGMA, THETA)
 -- statistics: R = gprnd (K, SIGMA, THETA, ROWS)
 -- statistics: R = gprnd (K, SIGMA, THETA, ROWS, COLS, ...)
 -- statistics: R = gprnd (K, SIGMA, THETA, [SZ])

     Random arrays from the generalized Pareto distribution.

     ‘R = gprnd (K, SIGMA, THETA)’ returns an array of random numbers chosen
     from the generalized Pareto distribution with shape parameter K, scale
     parameter SIGMA, and location parameter THETA.  The size of R is the common
     size of K, SIGMA, and THETA.  A scalar input functions as a constant matrix
     of the same size as the other inputs.

     When called with a single size argument, ‘gprnd’ returns a square matrix
     with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     When K = 0 and THETA = 0, the Generalized Pareto is equivalent to the
     exponential distribution.  When K > 0 and ‘THETA = K / K’ the Generalized
     Pareto is equivalent to the Pareto distribution.  The mean of the
     Generalized Pareto is not finite when K >= 1 and the variance is not finite
     when K >= 1/2.  When K >= 0, the Generalized Pareto has positive density
     for X > THETA, or, when THETA < 0, for 0 <= (X - THETA) / SIGMA <= -1 / K.

     Further information about the generalized Pareto distribution can be found
     at <https://en.wikipedia.org/wiki/Generalized_Pareto_distribution>

     See also: gpcdf, gpinv, gppdf, gpfit, gplike, gpstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 55
Random arrays from the generalized Pareto distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 9
gumbelcdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 2184
 -- statistics: P = gumbelcdf (X)
 -- statistics: P = gumbelcdf (X, MU)
 -- statistics: P = gumbelcdf (X, MU, BETA)
 -- statistics: P = gumbelcdf (..., "upper")
 -- statistics: [P, PLO, PUP] = gumbelcdf (X, MU, BETA, PCOV)
 -- statistics: [P, PLO, PUP] = gumbelcdf (X, MU, BETA, PCOV, ALPHA)
 -- statistics: [P, PLO, PUP] = gumbelcdf (..., "upper")

     Gumbel cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the Gumbel distribution (also known as the extreme value or the type I
     generalized extreme value distribution) with location parameter MU and
     scale parameter BETA.  The size of P is the common size of X, MU and BETA.
     A scalar input functions as a constant matrix of the same size as the other
     inputs.

     Default values are MU = 0 and BETA = 1.

     When called with three output arguments, i.e.  ‘[P, PLO, PUP]’, ‘gumbelcdf’
     computes the confidence bounds for P when the input parameters MU and BETA
     are estimates.  In such case, PCOV, a 2x2 matrix containing the covariance
     matrix of the estimated parameters, is necessary.  Optionally, ALPHA, which
     has a default value of 0.05, specifies the 100 * (1 - ALPHA) percent
     confidence bounds.  PLO and PUP are arrays of the same size as P containing
     the lower and upper confidence bounds.

     ‘[...] = gumbelcdf (..., "upper")’ computes the upper tail probability of
     the Gumbel distribution with parameters MU and BETA, at the values in X.

     The Gumbel distribution is used to model the distribution of the maximum
     (or the minimum) of a number of samples of various distributions.  This
     version is suitable for modeling maxima.  For modeling minima, use the
     alternative extreme value CDF, ‘evcdf’.

     ‘[...] = gumbelcdf (..., "upper")’ computes the upper tail probability of
     the extreme value (Gumbel) distribution.

     Further information about the Gumbel distribution can be found at
     <https://en.wikipedia.org/wiki/Gumbel_distribution>

     See also: gumbelinv, gumbelpdf, gumbelrnd, gumbelfit, gumbellike,
     gumbelstat, evcdf.


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Gumbel cumulative distribution function (CDF).



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# elements: 1
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gumbelinv


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# type: sq_string
# elements: 1
# length: 1796
 -- statistics: X = gumbelinv (P)
 -- statistics: X = gumbelinv (P, MU)
 -- statistics: X = gumbelinv (P, MU, BETA)
 -- statistics: [X, XLO, XUP] = gumbelinv (P, MU, BETA, PCOV)
 -- statistics: [X, XLO, XUP] = gumbelinv (P, MU, BETA, PCOV, ALPHA)

     Inverse of the Gumbel cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     Gumbel distribution (also known as the extreme value or the type I
     generalized extreme value distribution) with location parameter MU and
     scale parameter BETA.  The size of X is the common size of P, MU and BETA.
     A scalar input functions as a constant matrix of the same size as the other
     inputs.

     Default values are MU = 0 and BETA = 1.

     When called with three output arguments, i.e.  [X, XLO, XUP], ‘gumbelinv’
     computes the confidence bounds for X when the input parameters MU and BETA
     are estimates.  In such case, PCOV, a 2x2 matrix containing the covariance
     matrix of the estimated parameters, is necessary.  Optionally, ALPHA, which
     has a default value of 0.05, specifies the 100 * (1 - ALPHA) percent
     confidence bounds.  XLO and XUP are arrays of the same size as X containing
     the lower and upper confidence bounds.

     The Gumbel distribution is used to model the distribution of the maximum
     (or the minimum) of a number of samples of various distributions.  This
     version is suitable for modeling maxima.  For modeling minima, use the
     alternative extreme value iCDF, ‘evinv’.

     Further information about the Gumbel distribution can be found at
     <https://en.wikipedia.org/wiki/Gumbel_distribution>

     See also: gumbelcdf, gumbelpdf, gumbelrnd, gumbelfit, gumbellike,
     gumbelstat, evinv.


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Inverse of the Gumbel cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 9
gumbelpdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1122
 -- statistics: Y = gumbelpdf (X)
 -- statistics: Y = gumbelpdf (X, MU)
 -- statistics: Y = gumbelpdf (X, MU, BETA)

     Gumbel probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the Gumbel distribution (also known as the extreme value or the type I
     generalized extreme value distribution) with location parameter MU and
     scale parameter BETA.  The size of Y is the common size of X, MU and BETA.
     A scalar input functions as a constant matrix of the same size as the other
     inputs.

     Default values are MU = 0 and BETA = 1.

     The Gumbel distribution is used to model the distribution of the maximum
     (or the minimum) of a number of samples of various distributions.  This
     version is suitable for modeling maxima.  For modeling minima, use the
     alternative extreme value iCDF, ‘evpdf’.

     Further information about the Gumbel distribution can be found at
     <https://en.wikipedia.org/wiki/Gumbel_distribution>

     See also: gumbelcdf, gumbelinv, gumbelrnd, gumbelfit, gumbellike,
     gumbelstat, evpdf.


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Gumbel probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 9
gumbelrnd


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# elements: 1
# length: 1543
 -- statistics: R = gumbelrnd (MU, BETA)
 -- statistics: R = gumbelrnd (MU, BETA, ROWS)
 -- statistics: R = gumbelrnd (MU, BETA, ROWS, COLS, ...)
 -- statistics: R = gumbelrnd (MU, BETA, [SZ])

     Random arrays from the Gumbel distribution.

     ‘R = gumbelrnd (MU, BETA)’ returns an array of random numbers chosen from
     the Gumbel distribution (also known as the extreme value or the type I
     generalized extreme value distribution) with location parameter MU and
     scale parameter BETA.  The size of R is the common size of MU and BETA.  A
     scalar input functions as a constant matrix of the same size as the other
     inputs.

     When called with a single size argument, ‘gumbelrnd’ returns a square
     matrix with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     The Gumbel distribution is used to model the distribution of the maximum
     (or the minimum) of a number of samples of various distributions.  This
     version is suitable for modeling maxima.  For modeling minima, use the
     alternative extreme value iCDF, ‘evinv’.

     Further information about the Gumbel distribution can be found at
     <https://en.wikipedia.org/wiki/Gumbel_distribution>

     See also: gumbelcdf, gumbelinv, gumbelpdf, gumbelfit, gumbellike,
     gumbelstat, evrnd.


# name: <cell-element>
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Random arrays from the Gumbel distribution.



# name: <cell-element>
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hncdf


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# length: 905
 -- statistics: P = hncdf (X, MU, SIGMA)
 -- statistics: P = hncdf (X, MU, SIGMA, "upper")

     Half-normal cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the half-normal distribution with location parameter MU and scale
     parameter SIGMA.  The size of P is the common size of X, MU and SIGMA.  A
     scalar input functions as a constant matrix of the same size as the other
     inputs.

     ‘[...] = hncdf (X, MU, SIGMA, "upper")’ computes the upper tail probability
     of the half-normal distribution with parameters MU and SIGMA, at the values
     in X.

     The half-normal CDF is only defined for X >= MU.

     Further information about the half-normal distribution can be found at
     <https://en.wikipedia.org/wiki/Half-normal_distribution>

     See also: hninv, hnpdf, hnrnd, hnfit, hnlike, hnstat.


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Half-normal cumulative distribution function (CDF).



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# elements: 1
# length: 5
hninv


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# type: sq_string
# elements: 1
# length: 629
 -- statistics: X = hninv (P, MU, SIGMA)

     Inverse of the half-normal cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     half-normal distribution with location parameter MU and scale parameter
     SIGMA.  The size of X is the common size of P, MU, and SIGMA.  A scalar
     input functions as a constant matrix of the same size as the other inputs.

     Further information about the half-normal distribution can be found at
     <https://en.wikipedia.org/wiki/Half-normal_distribution>

     See also: hncdf, hnpdf, hnrnd, hnfit, hnlike, hnstat.


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Inverse of the half-normal cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
hnpdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 665
 -- statistics: Y = hnpdf (X, MU, SIGMA)

     Half-normal probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the half-normal distribution with location parameter MU and scale parameter
     SIGMA.  The size of Y is the common size of X, MU, and SIGMA.  A scalar
     input functions as a constant matrix of the same size as the other inputs.

     The half-normal CDF is only defined for X >= MU.

     Further information about the half-normal distribution can be found at
     <https://en.wikipedia.org/wiki/Half-normal_distribution>

     See also: hncdf, hninv, hnrnd, hnfit, hnlike, hnstat.


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Half-normal probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
hnrnd


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# type: sq_string
# elements: 1
# length: 1131
 -- statistics: R = hnrnd (MU, SIGMA)
 -- statistics: R = hnrnd (MU, SIGMA, ROWS)
 -- statistics: R = hnrnd (MU, SIGMA, ROWS, COLS, ...)
 -- statistics: R = hnrnd (MU, SIGMA, [SZ])

     Random arrays from the half-normal distribution.

     ‘R = hnrnd (MU, SIGMA)’ returns an array of random numbers chosen from the
     half-normal distribution with location parameter MU and scale parameter
     SIGMA.  The size of R is the common size of MU and SIGMA.  A scalar input
     functions as a constant matrix of the same size as the other inputs.

     When called with a single size argument, ‘hnrnd’ returns a square matrix
     with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     Further information about the half-normal distribution can be found at
     <https://en.wikipedia.org/wiki/Half-normal_distribution>

     See also: hncdf, hninv, hnpdf, hnfit, hnlike, hnstat.


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Random arrays from the half-normal distribution.



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# elements: 1
# length: 7
hygecdf


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# length: 1133
 -- statistics: P = hygecdf (X, M, K, N)
 -- statistics: P = hygecdf (X, M, K, N, "upper")

     Hypergeometric cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the hypergeometric distribution with parameters M, K, and N.  The size
     of P is the common size of X, M, K, and N.  A scalar input functions as a
     constant matrix of the same size as the other inputs.

     This is the cumulative probability of obtaining not more than X marked
     items when randomly drawing a sample of size N without replacement from a
     population of total size M containing K marked items.  The parameters M, K,
     and N must be positive integers with K and N not greater than M.

     ‘[...] = hygecdf (X, M, K, N, "upper")’ computes the upper tail probability
     of the hypergeometric distribution with parameters M, K, and N, at the
     values in X.

     Further information about the hypergeometric distribution can be found at
     <https://en.wikipedia.org/wiki/Hypergeometric_distribution>

     See also: hygeinv, hygepdf, hygernd, hygestat.


# name: <cell-element>
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Hypergeometric cumulative distribution function (CDF).



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# elements: 1
# length: 7
hygeinv


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# type: sq_string
# elements: 1
# length: 903
 -- statistics: X = hygeinv (P, M, K, N)

     Inverse of the hypergeometric cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     hypergeometric distribution with parameters M, K, and N.  The size of X is
     the common size of P, M, K, and N.  A scalar input functions as a constant
     matrix of the same size as the other inputs.

     This is the number of drawn marked items X given a probability P, when
     randomly drawing a sample of size N without replacement from a population
     of total size M containing K marked items.  The parameters M, K, and N must
     be positive integers with K and N not greater than M.

     Further information about the hypergeometric distribution can be found at
     <https://en.wikipedia.org/wiki/Hypergeometric_distribution>

     See also: hygeinv, hygepdf, hygernd, hygestat.


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Inverse of the hypergeometric cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
hygepdf


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# elements: 1
# length: 1301
 -- statistics: Y = hygepdf (X, M, K, N)
 -- statistics: Y = hygepdf (..., "vectorexpand")

     Hypergeometric probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the hypergeometric distribution with parameters M, K, and N.  The size of Y
     is the common size of X, M, K, and N.  A scalar input functions as a
     constant matrix of the same size as the other inputs.

     This is the probability of obtaining X marked items when randomly drawing a
     sample of size N without replacement from a population of total size M
     containing K marked items.  The parameters M, K, and N must be positive
     integers with K and N not greater than M.

     If the optional parameter vectorexpand is provided, X may be an array with
     size different from parameters M, K, and N (which must still be of a common
     size or scalar).  Each element of X will be evaluated against each set of
     parameters M, K, and N in columnwise order.  The output Y will be an array
     of size R x S, where R = numel (M), and S = numel (X).

     Further information about the hypergeometric distribution can be found at
     <https://en.wikipedia.org/wiki/Hypergeometric_distribution>

     See also: hygecdf, hygeinv, hygernd, hygestat.


# name: <cell-element>
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# length: 50
Hypergeometric probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
hygernd


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# length: 1209
 -- statistics: R = hygernd (M, K, N)
 -- statistics: R = hygernd (M, K, N, ROWS)
 -- statistics: R = hygernd (M, K, N, ROWS, COLS, ...)
 -- statistics: R = hygernd (M, K, N, [SZ])

     Random arrays from the hypergeometric distribution.

     ‘R = hygernd ((M, K, N’ returns an array of random numbers chosen from the
     hypergeometric distribution with parameters M, K, and N.  The size of R is
     the common size of M, K, and N.  A scalar input functions as a constant
     matrix of the same size as the other inputs.

     The parameters M, K, and N must be positive integers with K and N not
     greater than M.

     When called with a single size argument, ‘hygernd’ returns a square matrix
     with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     Further information about the hypergeometric distribution can be found at
     <https://en.wikipedia.org/wiki/Hypergeometric_distribution>

     See also: hygecdf, hygeinv, hygepdf, hygestat.


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Random arrays from the hypergeometric distribution.



# name: <cell-element>
# type: sq_string
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# length: 7
invgcdf


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# length: 966
 -- statistics: P = invgcdf (X, MU, LAMBDA)
 -- statistics: P = invgcdf (X, MU, LAMBDA, "upper")

     Inverse Gaussian cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the inverse Gaussian distribution with scale parameter MU and shape
     parameter LAMBDA.  The size of P is the common size of X, MU and LAMBDA.  A
     scalar input functions as a constant matrix of the same size as the other
     inputs.

     ‘P = invgcdf (X, MU, LAMBDA, "upper")’ computes the upper tail probability
     of the inverse Gaussian distribution with parameters MU and LAMBDA, at the
     values in X.

     The inverse Gaussian CDF is only defined for MU > 0 and LAMBDA > 0.

     Further information about the inverse Gaussian distribution can be found at
     <https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution>

     See also: invginv, invgpdf, invgrnd, invgfit, invglike, invgstat.


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Inverse Gaussian cumulative distribution function (CDF).



# name: <cell-element>
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# elements: 1
# length: 7
invginv


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# type: sq_string
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# length: 737
 -- statistics: X = invginv (P, MU, LAMBDA)

     Inverse of the inverse Gaussian cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     inverse Gaussian distribution with scale parameter MU and shape parameter
     LAMBDA.  The size of X is the common size of P, MU, and LAMBDA.  A scalar
     input functions as a constant matrix of the same size as the other inputs.

     The inverse Gaussian CDF is only defined for MU > 0 and LAMBDA > 0.

     Further information about the inverse Gaussian distribution can be found at
     <https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution>

     See also: invgcdf, invgpdf, invgrnd, invgfit, invglike, invgstat.


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Inverse of the inverse Gaussian cumulative distribution function (iCDF).



# name: <cell-element>
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# elements: 1
# length: 7
invgpdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 722
 -- statistics: Y = invgpdf (X, MU, LAMBDA)

     Inverse Gaussian probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the inverse Gaussian distribution with scale parameter MU and shape
     parameter LAMBDA.  The size of Y is the common size of X, MU, and LAMBDA.
     A scalar input functions as a constant matrix of the same size as the other
     inputs.

     The inverse Gaussian CDF is only defined for MU > 0 and LAMBDA > 0.

     Further information about the inverse Gaussian distribution can be found at
     <https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution>

     See also: invgcdf, invginv, invgrnd, invgfit, invglike, invgstat.


# name: <cell-element>
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Inverse Gaussian probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
invgrnd


# name: <cell-element>
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# length: 1261
 -- statistics: R = invgrnd (MU, LAMBDA)
 -- statistics: R = invgrnd (MU, LAMBDA, ROWS)
 -- statistics: R = invgrnd (MU, LAMBDA, ROWS, COLS, ...)
 -- statistics: R = invgrnd (MU, LAMBDA, [SZ])

     Random arrays from the inverse Gaussian distribution.

     ‘R = invgrnd (MU, LAMBDA)’ returns an array of random numbers chosen from
     the inverse Gaussian distribution with location parameter MU and scale
     parameter LAMBDA.  The size of R is the common size of MU and LAMBDA.  A
     scalar input functions as a constant matrix of the same size as the other
     inputs.

     When called with a single size argument, ‘invgrnd’ returns a square matrix
     with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     The inverse Gaussian CDF is only defined for MU > 0 and LAMBDA > 0.

     Further information about the inverse Gaussian distribution can be found at
     <https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution>

     See also: invgcdf, invginv, invgpdf, invgfit, invglike, invgstat.


# name: <cell-element>
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# length: 53
Random arrays from the inverse Gaussian distribution.



# name: <cell-element>
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# elements: 1
# length: 8
iwishpdf


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 -- statistics: Y = iwishpdf (W, TAU, DF, LOG_Y=false)

     Compute the probability density function of the inverse Wishart
     distribution.

     Inputs: A P x P matrix W where to find the PDF and the P x P positive
     definite scale matrix TAU and scalar degrees of freedom parameter DF
     characterizing the inverse Wishart distribution.  (For the density to be
     finite, need DF > (P - 1).)  If the flag LOG_Y is set, return the log
     probability density - this helps avoid underflow when the numerical value
     of the density is very small.

     Output: Y is the probability density of Wishart(SIGMA, DF) at W.

     See also: iwishrnd, wishpdf, wishrnd.


# name: <cell-element>
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Compute the probability density function of the inverse Wishart distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
iwishrnd


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# type: sq_string
# elements: 1
# length: 1085
 -- statistics: [W, DI] = iwishrnd (TAU, DF, DI, N=1)

     Return a random matrix sampled from the inverse Wishart distribution with
     given parameters.

     Inputs: the p x p positive definite matrix TAU and scalar degrees of
     freedom parameter DF (and optionally the transposed Cholesky factor DI of
     SIGMA = ‘inv(Tau)’).

     DF can be non-integer as long as DF > d

     Output: a random p x p matrix W from the inverse Wishart(TAU, DF)
     distribution.  (‘inv(W)’ is from the Wishart(‘inv(Tau)’, DF) distribution.)
     If N > 1, then W is P x P x N and holds N such random matrices.
     (Optionally, the transposed Cholesky factor DI of SIGMA is also returned.)

     Averaged across many samples, the mean of W should approach TAU / (DF - P -
     1).

     References
     ----------

       1. Yu-Cheng Ku and Peter Bloomfield (2010), Generating Random Wishart
          Matrices with Fractional Degrees of Freedom in OX,
          http://www.gwu.edu/~forcpgm/YuChengKu-030510final-WishartYu-ChengKu.pdf

     See also: iwishpdf, wishpdf, wishrnd.


# name: <cell-element>
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Return a random matrix sampled from the inverse Wishart distribution with giv...



# name: <cell-element>
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jsucdf


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# type: sq_string
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# length: 580
 -- statistics: P = jsucdf (X)
 -- statistics: P = jsucdf (X, ALPHA1)
 -- statistics: P = jsucdf (X, ALPHA1, ALPHA2)

     Johnson SU cumulative distribution function (CDF).

     For each element of X, return the cumulative distribution functions (CDF)
     at X of the Johnson SU distribution with shape parameters ALPHA1 and
     ALPHA2.  The size of P is the common size of the input arguments X, ALPHA1,
     and ALPHA2.  A scalar input functions as a constant matrix of the same size
     as the other

     Default values are ALPHA1 = 1, ALPHA2 = 1.

     See also: jsupdf.


# name: <cell-element>
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Johnson SU cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
jsupdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 571
 -- statistics: Y = jsupdf (X)
 -- statistics: Y = jsupdf (X, ALPHA1)
 -- statistics: Y = jsupdf (X, ALPHA1, ALPHA2)

     Johnson SU probability density function (PDF).

     For each element of X, compute the probability density function (PDF) at X
     of the Johnson SU distribution with shape parameters ALPHA1 and ALPHA2.
     The size of P is the common size of the input arguments X, ALPHA1, and
     ALPHA2.  A scalar input functions as a constant matrix of the same size as
     the other

     Default values are ALPHA1 = 1, ALPHA2 = 1.

     See also: jsucdf.


# name: <cell-element>
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Johnson SU probability density function (PDF).



# name: <cell-element>
# type: sq_string
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laplacecdf


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# length: 936
 -- statistics: P = laplacecdf (X, MU, BETA)
 -- statistics: P = laplacecdf (X, MU, BETA, "upper")

     Laplace cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the Laplace distribution with location parameter MU and scale parameter
     (i.e.  "diversity") BETA.  The size of P is the common size of X, MU, and
     BETA.  A scalar input functions as a constant matrix of the same size as
     the other inputs.

     Both parameters must be reals and BETA > 0.  For BETA <= 0, NaN is
     returned.

     ‘P = laplacecdf (X, MU, BETA, "upper")’ computes the upper tail probability
     of the Laplace distribution with parameters MU and BETA, at the values in
     X.

     Further information about the Laplace distribution can be found at
     <https://en.wikipedia.org/wiki/Laplace_distribution>

     See also: laplaceinv, laplacepdf, laplacernd.


# name: <cell-element>
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Laplace cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 10
laplaceinv


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# type: sq_string
# elements: 1
# length: 719
 -- statistics: X = laplaceinv (P, MU, BETA)

     Inverse of the Laplace cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     Laplace distribution with location parameter MU and scale parameter (i.e.
     "diversity") BETA.  The size of X is the common size of P, MU, and BETA.  A
     scalar input functions as a constant matrix of the same size as the other
     inputs.

     Both parameters must be reals and BETA > 0.  For BETA <= 0, NaN is
     returned.

     Further information about the Laplace distribution can be found at
     <https://en.wikipedia.org/wiki/Laplace_distribution>

     See also: laplaceinv, laplacepdf, laplacernd.


# name: <cell-element>
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Inverse of the Laplace cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 10
laplacepdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 701
 -- statistics: Y = laplacepdf (X, MU, BETA)

     Laplace probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the Laplace distribution with location parameter MU and scale parameter
     (i.e.  "diversity") BETA.  The size of Y is the common size of X, MU, and
     BETA.  A scalar input functions as a constant matrix of the same size as
     the other inputs.

     Both parameters must be reals and BETA > 0.  For BETA <= 0, NaN is
     returned.

     Further information about the Laplace distribution can be found at
     <https://en.wikipedia.org/wiki/Laplace_distribution>

     See also: laplacecdf, laplacepdf, laplacernd.


# name: <cell-element>
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Laplace probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 10
laplacernd


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# type: sq_string
# elements: 1
# length: 1218
 -- statistics: R = laplacernd (MU, BETA)
 -- statistics: R = laplacernd (MU, BETA, ROWS)
 -- statistics: R = laplacernd (MU, BETA, ROWS, COLS, ...)
 -- statistics: R = laplacernd (MU, BETA, [SZ])

     Random arrays from the Laplace distribution.

     ‘R = laplacernd (MU, BETA)’ returns an array of random numbers chosen from
     the Laplace distribution with location parameter MU and scale parameter
     BETA.  The size of R is the common size of MU and BETA.  A scalar input
     functions as a constant matrix of the same size as the other inputs.

     Both parameters must be reals and BETA > 0.  For BETA <= 0, NaN is
     returned.

     When called with a single size argument, ‘laplacernd’ returns a square
     matrix with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     Further information about the Laplace distribution can be found at
     <https://en.wikipedia.org/wiki/Laplace_distribution>

     See also: laplacecdf, laplaceinv, laplacernd.


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Random arrays from the Laplace distribution.



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logicdf


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# length: 935
 -- statistics: P = logicdf (X, MU, SIGMA)
 -- statistics: P = logicdf (X, MU, SIGMA, "upper")

     Logistic cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the logistic distribution with location parameter MU and scale parameter
     SIGMA.  The size of P is the common size of X, MU, and SIGMA.  A scalar
     input functions as a constant matrix of the same size as the other inputs.

     Both parameters must be reals and SIGMA > 0.  For SIGMA <= 0, NaN is
     returned.

     ‘P = logicdf (X, MU, SIGMA, "upper")’ computes the upper tail probability
     of the logistic distribution with parameters MU and SIGMA, at the values in
     X.

     Further information about the logistic distribution can be found at
     <https://en.wikipedia.org/wiki/Logistic_distribution>

     See also: logiinv, logipdf, logirnd, logifit, logilike, logistat.


# name: <cell-element>
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Logistic cumulative distribution function (CDF).



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logiinv


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# type: sq_string
# elements: 1
# length: 720
 -- statistics: X = logiinv (P, MU, SIGMA)

     Inverse of the logistic cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     logistic distribution with location parameter MU and scale parameter SIGMA.
     The size of P is the common size of X, MU, and SIGMA.  A scalar input
     functions as a constant matrix of the same size as the other inputs.

     Both parameters must be reals and SIGMA > 0.  For SIGMA <= 0, NaN is
     returned.

     Further information about the logistic distribution can be found at
     <https://en.wikipedia.org/wiki/Logistic_distribution>

     See also: logicdf, logipdf, logirnd, logifit, logilike, logistat.


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Inverse of the logistic cumulative distribution function (iCDF).



# name: <cell-element>
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# length: 7
logipdf


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# length: 702
 -- statistics: Y = logipdf (X, MU, SIGMA)

     Logistic probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the logistic distribution with location parameter MU and scale parameter
     SIGMA.  The size of P is the common size of X, MU, and SIGMA.  A scalar
     input functions as a constant matrix of the same size as the other inputs.

     Both parameters must be reals and SIGMA > 0.  For SIGMA <= 0, NaN is
     returned.

     Further information about the logistic distribution can be found at
     <https://en.wikipedia.org/wiki/Logistic_distribution>

     See also: logicdf, logiinv, logirnd, logifit, logilike, logistat.


# name: <cell-element>
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Logistic probability density function (PDF).



# name: <cell-element>
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# elements: 1
# length: 7
logirnd


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# length: 1232
 -- statistics: R = logirnd (MU, SIGMA)
 -- statistics: R = logirnd (MU, SIGMA, ROWS)
 -- statistics: R = logirnd (MU, SIGMA, ROWS, COLS, ...)
 -- statistics: R = logirnd (MU, SIGMA, [SZ])

     Random arrays from the logistic distribution.

     ‘R = logirnd (MU, SIGMA)’ returns an array of random numbers chosen from
     the logistic distribution with location parameter MU and scale parameter
     SIGMA.  The size of R is the common size of MU and SIGMA.  A scalar input
     functions as a constant matrix of the same size as the other inputs.

     Both parameters must be reals and SIGMA > 0.  For SIGMA <= 0, NaN is
     returned.

     When called with a single size argument, ‘logirnd’ returns a square matrix
     with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     Further information about the logistic distribution can be found at
     <https://en.wikipedia.org/wiki/Logistic_distribution>

     See also: logcdf, logiinv, logipdf, logifit, logilike, logistat.


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Random arrays from the logistic distribution.



# name: <cell-element>
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loglcdf


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 -- statistics: P = loglcdf (X, MU, SIGMA)
 -- statistics: P = loglcdf (X, MU, SIGMA, "upper")

     Loglogistic cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the loglogistic distribution with mean parameter MU and scale parameter
     SIGMA.  The size of P is the common size of X, MU, and SIGMA.  A scalar
     input functions as a constant matrix of the same size as the other inputs.

     Mean of logarithmic values MU must be a non-negative real value, scale
     parameter of logarithmic values SIGMA must be a positive real value and X
     is supported in the range [0,Inf), otherwise NaN is returned.

     ‘P = loglcdf (X, MU, SIGMA, "upper")’ computes the upper tail probability
     of the log-logistic distribution with parameters MU and SIGMA, at the
     values in X.

     Further information about the loglogistic distribution can be found at
     <https://en.wikipedia.org/wiki/Log-logistic_distribution>

     OCTAVE/MATLAB use an alternative parameterization given by the pair μ, σ,
     i.e.  MU and SIGMA, in analogy with the logistic distribution.  Their
     relation to the α and b parameters used in Wikipedia are given below:

        • MU = log (A)
        • SIGMA = 1 / A

     See also: loglinv, loglpdf, loglrnd, loglfit, logllike, loglstat.


# name: <cell-element>
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Loglogistic cumulative distribution function (CDF).



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# length: 7
loglinv


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# length: 1147
 -- statistics: X = loglinv (P, MU, SIGMA)

     Inverse of the log-logistic cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     log-logistic distribution with mean parameter MU and scale parameter SIGMA.
     The size of X is the common size of P, MU, and SIGMA.  A scalar input
     functions as a constant matrix of the same size as the other inputs.

     Mean of logarithmic values MU must be a non-negative real value, scale
     parameter of logarithmic values SIGMA must be a positive real value and P
     is supported in the range [0,1], otherwise NaN is returned.

     Further information about the loglogistic distribution can be found at
     <https://en.wikipedia.org/wiki/Log-logistic_distribution>

     OCTAVE/MATLAB use an alternative parameterization given by the pair μ, σ,
     i.e.  MU and SIGMA, in analogy with the logistic distribution.  Their
     relation to the α and b parameters used in Wikipedia are given below:

        • MU = log (A)
        • SIGMA = 1 / A

     See also: loglcdf, loglpdf, loglrnd, loglfit, logllike, loglstat.


# name: <cell-element>
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Inverse of the log-logistic cumulative distribution function (iCDF).



# name: <cell-element>
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# elements: 1
# length: 7
loglpdf


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# length: 1127
 -- statistics: Y = loglpdf (X, MU, SIGMA)

     Loglogistic probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the loglogistic distribution with mean parameter MU and scale parameter
     SIGMA.  The size of Y is the common size of X, MU, and SIGMA.  A scalar
     input functions as a constant matrix of the same size as the other inputs.

     Mean of logarithmic values MU must be a non-negative real value, scale
     parameter of logarithmic values SIGMA must be a positive real value and X
     is supported in the range [0,Inf), otherwise 0 is returned.

     Further information about the loglogistic distribution can be found at
     <https://en.wikipedia.org/wiki/Log-logistic_distribution>

     OCTAVE/MATLAB use an alternative parameterization given by the pair μ, σ,
     i.e.  MU and SIGMA, in analogy with the logistic distribution.  Their
     relation to the α and b parameters used in Wikipedia are given below:

        • MU = log (A)
        • SIGMA = 1 / A

     See also: loglcdf, loglinv, loglrnd, loglfit, logllike, loglstat.


# name: <cell-element>
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Loglogistic probability density function (PDF).



# name: <cell-element>
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# length: 7
loglrnd


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# elements: 1
# length: 1594
 -- statistics: R = loglrnd (MU, SIGMA)
 -- statistics: R = loglrnd (MU, SIGMA, ROWS)
 -- statistics: R = loglrnd (MU, SIGMA, ROWS, COLS, ...)
 -- statistics: R = loglrnd (MU, SIGMA, [SZ])

     Random arrays from the loglogistic distribution.

     ‘R = loglrnd (MU, SIGMA)’ returns an array of random numbers chosen from
     the loglogistic distribution with mean parameter MU and scale parameter
     SIGMA.  The size of R is the common size of MU and SIGMA.  A scalar input
     functions as a constant matrix of the same size as the other inputs.

     Mean of logarithmic values MU must be a non-negative real value and scale
     parameter of logarithmic values SIGMA must be a positive real value.

     When called with mu single size argument, ‘loglrnd’ returns mu square
     matrix with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with mu row vector of dimensions, SZ.

     Further information about the loglogistic distribution can be found at
     <https://en.wikipedia.org/wiki/Log-logistic_distribution>

     OCTAVE/MATLAB use an alternative parameterization given by the pair μ, σ,
     i.e.  MU and SIGMA, in analogy with the logistic distribution.  Their
     relation to the α and b parameters used in Wikipedia are given below:

        • MU = log (A)
        • SIGMA = 1 / A

     See also: loglcdf, loglinv, loglpdf, loglfit, logllike, loglstat.


# name: <cell-element>
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Random arrays from the loglogistic distribution.



# name: <cell-element>
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logncdf


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# length: 1961
 -- statistics: P = logncdf (X)
 -- statistics: P = logncdf (X, MU)
 -- statistics: P = logncdf (X, MU, SIGMA)
 -- statistics: P = logncdf (..., "upper")
 -- statistics: [P, PLO, PUP] = logncdf (X, MU, SIGMA, PCOV)
 -- statistics: [P, PLO, PUP] = logncdf (X, MU, SIGMA, PCOV, ALPHA)
 -- statistics: [P, PLO, PUP] = logncdf (..., "upper")

     Lognormal cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the lognormal distribution with mean parameter MU and standard deviation
     parameter SIGMA, each corresponding to the associated normal distribution.
     The size of P is the common size of X, MU, and SIGMA.  A scalar input
     functions as a constant matrix of the same size as the other inputs.

     If a random variable follows this distribution, its logarithm is normally
     distributed with mean MU and standard deviation SIGMA.

     Default parameter values are MU = 0 and SIGMA = 1.  Both parameters must be
     reals and SIGMA > 0.  For SIGMA <= 0, NaN is returned.

     When called with three output arguments, i.e.  [P, PLO, PUP], ‘logncdf’
     computes the confidence bounds for P when the input parameters MU and SIGMA
     are estimates.  In such case, PCOV, a 2x2 matrix containing the covariance
     matrix of the estimated parameters, is necessary.  Optionally, ALPHA, which
     has a default value of 0.05, specifies the 100 * (1 - ALPHA) percent
     confidence bounds.  PLO and PUP are arrays of the same size as P containing
     the lower and upper confidence bounds.

     ‘[...] = logncdf (..., "upper")’ computes the upper tail probability of the
     log-normal distribution with parameters MU and SIGMA, at the values in X.

     Further information about the lognormal distribution can be found at
     <https://en.wikipedia.org/wiki/Log-normal_distribution>

     See also: logninv, lognpdf, lognrnd, lognfit, lognlike, lognstat.


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Lognormal cumulative distribution function (CDF).



# name: <cell-element>
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# elements: 1
# length: 7
logninv


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# length: 1057
 -- statistics: X = logninv (P)
 -- statistics: X = logninv (P, MU)
 -- statistics: X = logninv (P, MU, SIGMA)

     Inverse of the lognormal cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     lognormal distribution with mean parameter MU and standard deviation
     parameter SIGMA, each corresponding to the associated normal distribution.
     The size of X is the common size of P, MU, and SIGMA.  A scalar input
     functions as a constant matrix of the same size as the other inputs.

     If a random variable follows this distribution, its logarithm is normally
     distributed with mean MU and standard deviation SIGMA.

     Default parameter values are MU = 0 and SIGMA = 1.  Both parameters must be
     reals and SIGMA > 0.  For SIGMA <= 0, NaN is returned.

     Further information about the lognormal distribution can be found at
     <https://en.wikipedia.org/wiki/Log-normal_distribution>

     See also: logncdf, lognpdf, lognrnd, lognfit, lognlike, lognstat.


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Inverse of the lognormal cumulative distribution function (iCDF).



# name: <cell-element>
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# elements: 1
# length: 7
lognpdf


# name: <cell-element>
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# length: 1038
 -- statistics: Y = lognpdf (X)
 -- statistics: Y = lognpdf (X, MU)
 -- statistics: Y = lognpdf (X, MU, SIGMA)

     Lognormal probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the lognormal distribution with mean parameter MU and standard deviation
     parameter SIGMA, each corresponding to the associated normal distribution.
     The size of Y is the common size of P, MU, and SIGMA.  A scalar input
     functions as a constant matrix of the same size as the other inputs.

     If a random variable follows this distribution, its logarithm is normally
     distributed with mean MU and standard deviation SIGMA.

     Default parameter values are MU = 0 and SIGMA = 1.  Both parameters must be
     reals and SIGMA > 0.  For SIGMA <= 0, NaN is returned.

     Further information about the lognormal distribution can be found at
     <https://en.wikipedia.org/wiki/Log-normal_distribution>

     See also: logncdf, logninv, lognrnd, lognfit, lognlike, lognstat.


# name: <cell-element>
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Lognormal probability density function (PDF).



# name: <cell-element>
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lognrnd


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# length: 1395
 -- statistics: R = lognrnd (MU, SIGMA)
 -- statistics: R = lognrnd (MU, SIGMA, ROWS)
 -- statistics: R = lognrnd (MU, SIGMA, ROWS, COLS, ...)
 -- statistics: R = lognrnd (MU, SIGMA, [SZ])

     Random arrays from the lognormal distribution.

     ‘R = lognrnd (MU, SIGMA)’ returns an array of random numbers chosen from
     the lognormal distribution with mean parameter MU and standard deviation
     parameter SIGMA, each corresponding to the associated normal distribution.
     The size of R is the common size of MU, and SIGMA.  A scalar input
     functions as a constant matrix of the same size as the other inputs.  Both
     parameters must be reals and SIGMA > 0.  For SIGMA <= 0, NaN is returned.

     Both parameters must be reals and SIGMA > 0.  For SIGMA <= 0, NaN is
     returned.

     When called with a single size argument, ‘lognrnd’ returns a square matrix
     with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     Further information about the lognormal distribution can be found at
     <https://en.wikipedia.org/wiki/Log-normal_distribution>

     See also: logncdf, logninv, lognpdf, lognfit, lognlike, lognstat.


# name: <cell-element>
# type: sq_string
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# length: 46
Random arrays from the lognormal distribution.



# name: <cell-element>
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# elements: 1
# length: 5
mnpdf


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# length: 1690
 -- statistics: Y = mnpdf (X, PK)

     Multinomial probability density function (PDF).

     Arguments
     ---------

        • X is vector with a single sample of a multinomial distribution with
          parameter PK or a matrix of random samples from multinomial
          distributions.  In the latter case, each row of X is a sample from a
          multinomial distribution with the corresponding row of PK being its
          parameter.

        • PK is a vector with the probabilities of the categories or a matrix
          with each row containing the probabilities of a multinomial sample.

     Return values
     -------------

        • Y is a vector of probabilities of the random samples X from the
          multinomial distribution with corresponding parameter PK.  The
          parameter N of the multinomial distribution is the sum of the elements
          of each row of X.  The length of Y is the number of columns of X.  If
          a row of PK does not sum to ‘1’, then the corresponding element of Y
          will be ‘NaN’.

     Examples
     --------

          x = [1, 4, 2];
          pk = [0.2, 0.5, 0.3];
          y = mnpdf (x, pk);

          x = [1, 4, 2; 1, 0, 9];
          pk = [0.2, 0.5, 0.3; 0.1, 0.1, 0.8];
          y = mnpdf (x, pk);

     References
     ----------

       1. Wendy L. Martinez and Angel R. Martinez.  ‘Computational Statistics
          Handbook with MATLAB’. Appendix E, pages 547-557, Chapman & Hall/CRC,
          2001.

       2. Merran Evans, Nicholas Hastings and Brian Peacock.  ‘Statistical
          Distributions’.  pages 134-136, Wiley, New York, third edition, 2000.

     See also: mnrnd.


# name: <cell-element>
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Multinomial probability density function (PDF).



# name: <cell-element>
# type: sq_string
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# length: 5
mnrnd


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# length: 2232
 -- statistics: R = mnrnd (N, PK)
 -- statistics: R = mnrnd (N, PK, S)

     Random arrays from the multinomial distribution.

     Arguments
     ---------

        • N is the first parameter of the multinomial distribution.  N can be
          scalar or a vector containing the number of trials of each multinomial
          sample.  The elements of N must be non-negative integers.

        • PK is the second parameter of the multinomial distribution.  PK can be
          a vector with the probabilities of the categories or a matrix with
          each row containing the probabilities of a multinomial sample.  If PK
          has more than one row and N is non-scalar, then the number of rows of
          PK must match the number of elements of N.

        • S is the number of multinomial samples to be generated.  S must be a
          non-negative integer.  If S is specified, then N must be scalar and PK
          must be a vector.

     Return values
     -------------

        • R is a matrix of random samples from the multinomial distribution with
          corresponding parameters N and PK.  Each row corresponds to one
          multinomial sample.  The number of columns, therefore, corresponds to
          the number of columns of PK.  If S is not specified, then the number
          of rows of R is the maximum of the number of elements of N and the
          number of rows of PK.  If a row of PK does not sum to ‘1’, then the
          corresponding row of R will contain only ‘NaN’ values.

     Examples
     --------

          n = 10;
          pk = [0.2, 0.5, 0.3];
          r = mnrnd (n, pk);

          n = 10 * ones (3, 1);
          pk = [0.2, 0.5, 0.3];
          r = mnrnd (n, pk);

          n = (1:2)';
          pk = [0.2, 0.5, 0.3; 0.1, 0.1, 0.8];
          r = mnrnd (n, pk);

     References
     ----------

       1. Wendy L. Martinez and Angel R. Martinez.  ‘Computational Statistics
          Handbook with MATLAB’. Appendix E, pages 547-557, Chapman & Hall/CRC,
          2001.

       2. Merran Evans, Nicholas Hastings and Brian Peacock.  ‘Statistical
          Distributions’.  pages 134-136, Wiley, New York, third edition, 2000.

     See also: mnpdf.


# name: <cell-element>
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# length: 48
Random arrays from the multinomial distribution.



# name: <cell-element>
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# length: 6
mvncdf


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 -- statistics: P = mvncdf (X)
 -- statistics: P = mvncdf (X, MU, SIGMA)
 -- statistics: P = mvncdf (X_LO, X_UP, MU, SIGMA)
 -- statistics: P = mvncdf (..., OPTIONS)
 -- statistics: [P, ERR] = mvncdf (...)

     Multivariate normal cumulative distribution function (CDF).

     ‘P = mvncdf (X)’ returns the cumulative probability of the multivariate
     normal distribution evaluated at each row of X with zero mean and an
     identity covariance matrix.  The rows of matrix X correspond to
     observations and its columns to variables.  The return argument P is a
     column vector with the same number of rows as in X.

     ‘P = mvncdf (X, MU, SIGMA)’ returns cumulative probability of the
     multivariate normal distribution evaluated at each row of X with mean MU
     and a covariance matrix SIGMA.  MU can be either a scalar (the same of
     every variable) or a row vector with the same number of elements as the
     number of variables in X.  SIGMA covariance matrix may be specified a row
     vector if it only contains variances along its diagonal and zero
     covariances of the diagonal.  In such a case, the diagonal vector SIGMA
     must have the same number of elements as the number of variables (columns)
     in X.  If you only want to specify sigma, you can pass an empty matrix for
     MU.

     The multivariate normal cumulative probability at X is defined as the
     probability that a random vector V, distributed as multivariate normal,
     will fall within the semi-infinite rectangle with upper limits defined by
     X.
        • Pr{V(1)<=X(1), V(2)<=X(2), ... V(D)<=X(D)}.

     ‘P = mvncdf (X_LO, X_HI, MU, SIGMA)’ returns the multivariate normal
     cumulative probability evaluated over the rectangle (hyper-rectangle for
     multivariate data in X) with lower and upper limits defined by X_LO and
     X_HI, respectively.

     ‘[P, ERR] = mvncdf (...)’ also returns an error estimate ERR in P.

     ‘P = mvncdf (..., OPTIONS)’ specifies the structure, which controls
     specific parameters for the numerical integration used to compute P.  The
     required fields are:

     "TolFun"              Maximum absolute error tolerance.  Default is 1e-8 for D <
                           4, or 1e-4 for D >= 4.  Note that for bivariate normal
                           cdf, the Octave implementation has a precision of more
                           than 1e-10.
                           
     "MaxFunEvals"         Maximum number of integrand evaluations.  Default is 1e7
                           for D > 4.
                           
     "Display"             Display options.  Choices are "off" (default), "iter",
                           which shows the probability and estimated error at each
                           repetition, and "final", which shows the final probability
                           and related error after the integrand has converged
                           successfully.

     See also: bvncdf, mvnpdf, mvnrnd.


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Multivariate normal cumulative distribution function (CDF).



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mvnpdf


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 -- statistics: Y = mvnpdf (X, MU, SIGMA)

     Multivariate normal probability density function (PDF).

     ‘Y = mvnpdf (X)’ returns the probability density of the multivariate normal
     distribution with zero mean and identity covariance matrix, evaluated at
     each row of X.  Rows of the N-by-D matrix X correspond to observations
     orpoints, and columns correspond to variables or coordinates.  Y is an
     N-by-1 vector.

     ‘Y = mvnpdf (X, MU)’ returns the density of the multivariate normal
     distribution with mean MU and identity covariance matrix, evaluated at each
     row of X.  MU is a 1-by-D vector, or an N-by-D matrix, in which case the
     density is evaluated for each row of X with the corresponding row of MU.
     MU can also be a scalar value, which MVNPDF replicates to match the size of
     X.

     ‘Y = mvnpdf (X, MU, SIGMA)’ returns the density of the multivariate normal
     distribution with mean MU and covariance SIGMA, evaluated at each row of X.
     SIGMA is a D-by-D matrix, or an D-by-D-by-N array, in which case the
     density is evaluated for each row of X with the corresponding page of
     SIGMA, i.e., ‘mvnpdf’ computes Y(I) using X(I,:) and SIGMA(:,:,I).  If the
     covariance matrix is diagonal, containing variances along the diagonal and
     zero covariances off the diagonal, SIGMA may also be specified as a 1-by-D
     matrix or a 1-by-D-by-N array, containing just the diagonal.  Pass in the
     empty matrix for MU to use its default value when you want to only specify
     SIGMA.

     If X is a 1-by-D vector, ‘mvnpdf’ replicates it to match the leading
     dimension of MU or the trailing dimension of SIGMA.

     See also: mvncdf, mvnrnd.


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Multivariate normal probability density function (PDF).



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mvnrnd


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 -- statistics: R = mvnrnd (MU, SIGMA)
 -- statistics: R = mvnrnd (MU, SIGMA, N)
 -- statistics: R = mvnrnd (MU, SIGMA, N, T)
 -- statistics: [R, T] = mvnrnd (...)

     Random vectors from the multivariate normal distribution.

     ‘R = mvnrnd (MU, SIGMA)’ returns an N-by-D matrix R of random vectors
     chosen from the multivariate normal distribution with mean vector MU and
     covariance matrix SIGMA.  MU is an N-by-D matrix, and ‘mvnrnd’ generates
     each N of R using the corresponding N of MU.  SIGMA is a D-by-D symmetric
     positive semi-definite matrix, or a D-by-D-by-N array.  If SIGMA is an
     array, ‘mvnrnd’ generates each N of R using the corresponding page of
     SIGMA, i.e., ‘mvnrnd’ computes R(I,:) using MU(I,:) and SIGMA(:,:,I).  If
     the covariance matrix is diagonal, containing variances along the diagonal
     and zero covariances off the diagonal, SIGMA may also be specified as a
     1-by-D matrix or a 1-by-D-by-N array, containing just the diagonal.  If MU
     is a 1-by-D vector, ‘mvnrnd’ replicates it to match the trailing dimension
     of SIGMA.

     ‘R = mvnrnd (MU, SIGMA, N)’ returns a N-by-D matrix R of random vectors
     chosen from the multivariate normal distribution with 1-by-D mean vector
     MU, and D-by-D covariance matrix SIGMA.

     ‘R = mvnrnd (MU, SIGMA, N, T)’ supplies the Cholesky factor T of SIGMA, so
     that SIGMA(:,:,J) == T(:,:,J)'*T(:,:,J) if SIGMA is a 3D array or SIGMA ==
     T'*T if SIGMA is a matrix.  No error checking is done on T.

     ‘[R, T] = mvnrnd (...)’ returns the Cholesky factor T, so it can be re-used
     to make later calls more efficient, although there are greater efficiency
     gains when SIGMA can be specified as a diagonal instead.

     See also: mvncdf, mvnpdf.


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Random vectors from the multivariate normal distribution.



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mvtcdf


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 -- statistics: P = mvtcdf (X, RHO, DF)
 -- statistics: P = mvncdf (X_LO, X_UP, RHO, DF)
 -- statistics: P = mvncdf (..., OPTIONS)
 -- statistics: [P, ERR] = mvncdf (...)

     Multivariate Student's t cumulative distribution function (CDF).

     ‘P = mvtcdf (X, RHO, DF)’ returns the cumulative probability of the
     multivariate student's t distribution with correlation parameters RHO and
     degrees of freedom DF, evaluated at each row of X.  The rows of the NxD
     matrix X correspond to sample observations and its columns correspond to
     variables or coordinates.  The return argument P is a column vector with
     the same number of rows as in X.

     RHO is a symmetric, positive definite, DxD correlation matrix.  DF is a
     scalar or a vector with N elements.

     Note: ‘mvtcdf’ computes the CDF for the standard multivariate Student's t
     distribution, centered at the origin, with no scale parameters.  If RHO is
     a covariance matrix, i.e.  ‘diag(RHO)’ is not all ones, ‘mvtcdf’ rescales
     RHO to transform it to a correlation matrix.  ‘mvtcdf’ does not rescale X,
     though.

     The multivariate Student's t cumulative probability at X is defined as the
     probability that a random vector T, distributed as multivariate normal,
     will fall within the semi-infinite rectangle with upper limits defined by
     X.
        • Pr{T(1)<=X(1), T(2)<=X(2), ... T(D)<=X(D)}.

     ‘P = mvtcdf (X_LO, X_HI, RHO, DF)’ returns the multivariate Student's t
     cumulative probability evaluated over the rectangle (hyper-rectangle for
     multivariate data in X) with lower and upper limits defined by X_LO and
     X_HI, respectively.

     ‘[P, ERR] = mvtcdf (...)’ also returns an error estimate ERR in P.

     ‘P = mvtcdf (..., OPTIONS)’ specifies the structure, which controls
     specific parameters for the numerical integration used to compute P.  The
     required fields are:

     "TolFun"              Maximum absolute error tolerance.  Default is 1e-8 for D <
                           4, or 1e-4 for D >= 4.
                           
     "MaxFunEvals"         Maximum number of integrand evaluations when D >= 4.
                           Default is 1e7.  Ignored when D < 4.
                           
     "Display"             Display options.  Choices are "off" (default), "iter",
                           which shows the probability and estimated error at each
                           repetition, and "final", which shows the final probability
                           and related error after the integrand has converged
                           successfully.  Ignored when D < 4.

     See also: bvtcdf, mvtpdf, mvtrnd, mvtcdfqmc.


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Multivariate Student's t cumulative distribution function (CDF).



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mvtcdfqmc


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 -- statistics: P = mvtcdfqmc (A, B, RHO, DF)
 -- statistics: P = mvtcdfqmc (..., TOLFUN)
 -- statistics: P = mvtcdfqmc (..., TOLFUN, MAXFUNEVALS)
 -- statistics: P = mvtcdfqmc (..., TOLFUN, MAXFUNEVALS, DISPLAY)
 -- statistics: [P, ERR] = mvtcdfqmc (...)
 -- statistics: [P, ERR, FUNEVALS] = mvtcdfqmc (...)

     Quasi-Monte-Carlo computation of the multivariate Student's T CDF.

     The QMC multivariate Student's t distribution is evaluated between the
     lower limit A and upper limit B of the hyper-rectangle with a correlation
     matrix RHO and degrees of freedom DF.

     "TolFun"         -- Maximum absolute error tolerance.  Default is 1e-4.
     "MaxFunEvals"    -- Maximum number of integrand evaluations.  Default is 1e7
                      for D > 4.
     "Display"        -- Display options.  Choices are "off" (default), "iter",
                      which shows the probability and estimated error at each
                      repetition, and "final", which shows the final probability and
                      related error after the integrand has converged successfully.

     ‘[P, ERR, FUNEVALS] = mvtcdfqmc (...)’ returns the estimated probability,
     P, an estimate of the error, ERR, and the number of iterations until a
     successful convergence is met, unless the value in MAXFUNEVALS was reached.

     See also: mvtcdf, mvtpdf, mvtrnd.


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Quasi-Monte-Carlo computation of the multivariate Student's T CDF.



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mvtpdf


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 -- statistics: Y = mvtpdf (X, RHO, DF)

     Multivariate Student's t probability density function (PDF).

     Arguments
     ---------

        • X are the points at which to find the probability, where each row
          corresponds to an observation.  (NxD matrix)

        • RHO is the correlation matrix.  (DxD symmetric positive definite
          matrix)

        • DF is the degrees of freedom.  (scalar or vector of length N)

     The distribution is assumed to be centered (zero mean).

     Return values
     -------------

        • Y is the probability density for each row of X.  (Nx1 vector)

     Examples
     --------

          x = [1 2];
          rho = [1.0 0.5; 0.5 1.0];
          df = 4;
          y = mvtpdf (x, rho, df)

     References
     ----------

       1. Michael Roth, On the Multivariate t Distribution, Technical report
          from Automatic Control at Linkoepings universitet,
          <http://users.isy.liu.se/en/rt/roth/student.pdf>

     See also: mvtcdf, mvtcdfqmc, mvtrnd.


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Multivariate Student's t probability density function (PDF).



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mvtrnd


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 -- statistics: R = mvtrnd (RHO, DF)
 -- statistics: R = mvtrnd (RHO, DF, N)

     Random vectors from the multivariate Student's t distribution.

     Arguments
     ---------

        • RHO is the matrix of correlation coefficients.  If there are any
          non-unit diagonal elements then RHO will be normalized, so that the
          resulting covariance of the obtained samples R follows: ‘cov (r) =
          df/(df-2) * rho ./ (sqrt (diag (rho) * diag (rho)))’.  In order to
          obtain samples distributed according to a standard multivariate
          student's t-distribution, RHO must be equal to the identity matrix.
          To generate multivariate student's t-distribution samples R with
          arbitrary covariance matrix RHO, the following scaling might be used:
          ‘r = mvtrnd (rho, df, n) * diag (sqrt (diag (rho)))’.

        • DF is the degrees of freedom for the multivariate t-distribution.  DF
          must be a vector with the same number of elements as samples to be
          generated or be scalar.

        • N is the number of rows of the matrix to be generated.  N must be a
          non-negative integer and corresponds to the number of samples to be
          generated.

     Return values
     -------------

        • R is a matrix of random samples from the multivariate t-distribution
          with N row samples.

     Examples
     --------

          rho = [1, 0.5; 0.5, 1];
          df = 3;
          n = 10;
          r = mvtrnd (rho, df, n);

          rho = [1, 0.5; 0.5, 1];
          df = [2; 3];
          n = 2;
          r = mvtrnd (rho, df, 2);

     References
     ----------

       1. Wendy L. Martinez and Angel R. Martinez.  ‘Computational Statistics
          Handbook with MATLAB’. Appendix E, pages 547-557, Chapman & Hall/CRC,
          2001.

       2. Samuel Kotz and Saralees Nadarajah.  ‘Multivariate t Distributions and
          Their Applications’.  Cambridge University Press, Cambridge, 2004.

     See also: mvtcdf, mvtcdfqmc, mvtpdf.


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Random vectors from the multivariate Student's t distribution.



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nakacdf


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 -- statistics: P = nakacdf (X, MU, OMEGA)
 -- statistics: P = nakacdf (X, MU, OMEGA, "upper")

     Nakagami cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the Nakagami distribution with shape parameter MU and spread parameter
     OMEGA.  The size of P is the common size of X, MU, and OMEGA.  A scalar
     input functions as a constant matrix of the same size as the other inputs.

     Both parameters must be positive reals and MU >= 0.5.  For MU < 0.5 or
     OMEGA <= 0, NaN is returned.

     ‘P = nakacdf (X, MU, OMEGA, "upper")’ computes the upper tail probability
     of the Nakagami distribution with parameters MU and BETA, at the values in
     X.

     Further information about the Nakagami distribution can be found at
     <https://en.wikipedia.org/wiki/Nakagami_distribution>

     See also: nakainv, nakapdf, nakarnd, nakafit, nakalike, nakastat.


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Nakagami cumulative distribution function (CDF).



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nakainv


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 -- statistics: X = nakacdf (X, MU, OMEGA)

     Inverse of the Nakagami cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     Nakagami distribution with shape parameter MU and spread parameter OMEGA.
     The size of X is the common size of X, MU, and OMEGA.  A scalar input
     functions as a constant matrix of the same size as the other inputs.

     Both parameters must be positive reals and MU >= 0.5.  For MU < 0.5 or
     OMEGA <= 0, NaN is returned.

     Further information about the Nakagami distribution can be found at
     <https://en.wikipedia.org/wiki/Nakagami_distribution>

     See also: nakacdf, nakapdf, nakarnd, nakafit, nakalike, nakastat.


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Inverse of the Nakagami cumulative distribution function (iCDF).



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nakapdf


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 -- statistics: Y = nakapdf (X, MU, OMEGA)

     Nakagami probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the Nakagami distribution with shape parameter MU and spread parameter
     OMEGA.  The size of Y is the common size of X, MU, and OMEGA.  A scalar
     input functions as a constant matrix of the same size as the other inputs.

     Both parameters must be positive reals and MU >= 0.5.  For MU < 0.5 or
     OMEGA <= 0, NaN is returned.

     Further information about the Nakagami distribution can be found at
     <https://en.wikipedia.org/wiki/Nakagami_distribution>

     See also: nakacdf, nakapdf, nakarnd, nakafit, nakalike, nakastat.


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Nakagami probability density function (PDF).



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nakarnd


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 -- statistics: R = nakarnd (MU, OMEGA)
 -- statistics: R = nakarnd (MU, OMEGA, ROWS)
 -- statistics: R = nakarnd (MU, OMEGA, ROWS, COLS, ...)
 -- statistics: R = nakarnd (MU, OMEGA, [SZ])

     Random arrays from the Nakagami distribution.

     ‘R = nakarnd (MU, OMEGA)’ returns an array of random numbers chosen from
     the Nakagami distribution with shape parameter MU and spread parameter
     OMEGA.  The size of R is the common size of MU and OMEGA.  A scalar input
     functions as a constant matrix of the same size as the other inputs.

     Both parameters must be positive reals and MU >= 0.5.  For MU < 0.5 or
     OMEGA <= 0, NaN is returned.

     When called with a single size argument, ‘nakarnd’ returns a square matrix
     with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     Further information about the Nakagami distribution can be found at
     <https://en.wikipedia.org/wiki/Nakagami_distribution>

     See also: nakacdf, nakainv, nakapdf, nakafit, nakalike, nakastat.


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Random arrays from the Nakagami distribution.



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nbincdf


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 -- statistics: P = nbincdf (X, R, PS)
 -- statistics: P = nbincdf (X, R, PS, "upper")

     Negative binomial cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the negative binomial distribution with parameters R and PS, where R is
     the number of successes until the experiment is stopped and PS is the
     probability of success in each experiment, given the number of failures in
     X.  The size of P is the common size of X, R, and PS.  A scalar input
     functions as a constant matrix of the same size as the other inputs.

     The algorithm uses the cumulative sums of the binomial masses.

     ‘P = nbincdf (X, R, PS, "upper")’ computes the upper tail probability of
     the negative binomial distribution with parameters R and PS, at the values
     in X.

     When R is an integer, the negative binomial distribution is also known as
     the Pascal distribution and it models the number of failures in X before a
     specified number of successes is reached in a series of independent,
     identical trials.  Its parameters are the probability of success in a
     single trial, PS, and the number of successes, R.  A special case of the
     negative binomial distribution, when R = 1, is the geometric distribution,
     which models the number of failures before the first success.

     R can also have non-integer positive values, in which form the negative
     binomial distribution, also known as the Polya distribution, has no
     interpretation in terms of repeated trials, but, like the Poisson
     distribution, it is useful in modeling count data.  The negative binomial
     distribution is more general than the Poisson distribution because it has a
     variance that is greater than its mean, making it suitable for count data
     that do not meet the assumptions of the Poisson distribution.  In the
     limit, as R increases to infinity, the negative binomial distribution
     approaches the Poisson distribution.

     Further information about the negative binomial distribution can be found
     at <https://en.wikipedia.org/wiki/Negative_binomial_distribution>

     See also: nbininv, nbinpdf, nbinrnd, nbinfit, nbinlike, nbinstat.


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Negative binomial cumulative distribution function (CDF).



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nbininv


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 -- statistics: X = nbininv (P, R, PS)

     Inverse of the negative binomial cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     negative binomial distribution with parameters R and PS, where R is the
     number of successes until the experiment is stopped and PS is the
     probability of success in each experiment, given the probability in P.  The
     size of X is the common size of P, R, and PS.  A scalar input functions as
     a constant matrix of the same size as the other inputs.

     When R is an integer, the negative binomial distribution is also known as
     the Pascal distribution and it models the number of failures in X before a
     specified number of successes is reached in a series of independent,
     identical trials.  Its parameters are the probability of success in a
     single trial, PS, and the number of successes, R.  A special case of the
     negative binomial distribution, when R = 1, is the geometric distribution,
     which models the number of failures before the first success.

     R can also have non-integer positive values, in which form the negative
     binomial distribution, also known as the Polya distribution, has no
     interpretation in terms of repeated trials, but, like the Poisson
     distribution, it is useful in modeling count data.  The negative binomial
     distribution is more general than the Poisson distribution because it has a
     variance that is greater than its mean, making it suitable for count data
     that do not meet the assumptions of the Poisson distribution.  In the
     limit, as R increases to infinity, the negative binomial distribution
     approaches the Poisson distribution.

     Further information about the negative binomial distribution can be found
     at <https://en.wikipedia.org/wiki/Negative_binomial_distribution>

     See also: nbininv, nbinpdf, nbinrnd, nbinfit, nbinlike, nbinstat.


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Inverse of the negative binomial cumulative distribution function (iCDF).



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nbinpdf


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 -- statistics: Y = nbinpdf (X, R, PS)

     Negative binomial probability density function (PDF).

     For each element of X, compute the probability density function (PDF) at X
     of the negative binomial distribution with parameters R and PS, where R is
     the number of successes until the experiment is stopped and PS is the
     probability of success in each experiment, given the number of failures in
     X.  The size of Y is the common size of X, R, and PS.  A scalar input
     functions as a constant matrix of the same size as the other inputs.

     When R is an integer, the negative binomial distribution is also known as
     the Pascal distribution and it models the number of failures in X before a
     specified number of successes is reached in a series of independent,
     identical trials.  Its parameters are the probability of success in a
     single trial, PS, and the number of successes, R.  A special case of the
     negative binomial distribution, when R = 1, is the geometric distribution,
     which models the number of failures before the first success.

     R can also have non-integer positive values, in which form the negative
     binomial distribution, also known as the Polya distribution, has no
     interpretation in terms of repeated trials, but, like the Poisson
     distribution, it is useful in modeling count data.  The negative binomial
     distribution is more general than the Poisson distribution because it has a
     variance that is greater than its mean, making it suitable for count data
     that do not meet the assumptions of the Poisson distribution.  In the
     limit, as R increases to infinity, the negative binomial distribution
     approaches the Poisson distribution.

     Further information about the negative binomial distribution can be found
     at <https://en.wikipedia.org/wiki/Negative_binomial_distribution>

     See also: nbininv, nbininv, nbinrnd, nbinfit, nbinlike, nbinstat.


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Negative binomial probability density function (PDF).



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nbinrnd


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 -- statistics: RND = nbinrnd (R, PS)
 -- statistics: RND = nbinrnd (R, PS, ROWS)
 -- statistics: RND = nbinrnd (R, PS, ROWS, COLS, ...)
 -- statistics: RND = nbinrnd (R, PS, [SZ])

     Random arrays from the negative binomial distribution.

     ‘RND = nbinrnd (R, PS)’ returns an array of random numbers chosen from the
     negative binomial distribution with parameters R and PS, where R is the
     number of successes until the experiment is stopped and PS is the
     probability of success in each experiment, given the number of failures in
     X.  The size of RND is the common size of R and PS.  A scalar input
     functions as a constant matrix of the same size as the other inputs.

     When called with a single size argument, return a square matrix with the
     dimension specified.  When called with more than one scalar argument the
     first two arguments are taken as the number of rows and columns and any
     further arguments specify additional matrix dimensions.  The size may also
     be specified with a vector of dimensions SZ.

     When R is an integer, the negative binomial distribution is also known as
     the Pascal distribution and it models the number of failures in X before a
     specified number of successes is reached in a series of independent,
     identical trials.  Its parameters are the probability of success in a
     single trial, PS, and the number of successes, R.  A special case of the
     negative binomial distribution, when R = 1, is the geometric distribution,
     which models the number of failures before the first success.

     R can also have non-integer positive values, in which form the negative
     binomial distribution, also known as the Polya distribution, has no
     interpretation in terms of repeated trials, but, like the Poisson
     distribution, it is useful in modeling count data.  The negative binomial
     distribution is more general than the Poisson distribution because it has a
     variance that is greater than its mean, making it suitable for count data
     that do not meet the assumptions of the Poisson distribution.  In the
     limit, as R increases to infinity, the negative binomial distribution
     approaches the Poisson distribution.

     Further information about the negative binomial distribution can be found
     at <https://en.wikipedia.org/wiki/Negative_binomial_distribution>

     See also: nbininv, nbininv, nbinpdf, nbinfit, nbinlike, nbinstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 54
Random arrays from the negative binomial distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
ncfcdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 904
 -- statistics: P = ncfcdf (X, DF1, DF2, LAMBDA)
 -- statistics: P = ncfcdf (X, DF1, DF2, LAMBDA, "upper")

     Noncentral F-cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the noncentral F-distribution with DF1 and DF2 degrees of freedom and
     noncentrality parameter LAMBDA.  The size of P is the common size of X,
     DF1, DF2, and LAMBDA.  A scalar input functions as a constant matrix of the
     same size as the other inputs.

     ‘P = ncfcdf (X, DF1, DF2, LAMBDA, "upper")’ computes the upper tail
     probability of the noncentral F-distribution with parameters DF1, DF2, and
     LAMBDA, at the values in X.

     Further information about the noncentral F-distribution can be found at
     <https://en.wikipedia.org/wiki/Noncentral_F-distribution>

     See also: ncfinv, ncfpdf, ncfrnd, ncfstat, fcdf.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 52
Noncentral F-cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
ncfinv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 735
 -- statistics: X = ncfinv (P, DF1, DF2, LAMBDA)

     Inverse of the noncentral F-cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     noncentral F-distribution with DF1 and DF2 degrees of freedom and
     noncentrality parameter LAMBDA.  The size of X is the common size of P,
     DF1, DF2, and LAMBDA.  A scalar input functions as a constant matrix of the
     same size as the other inputs.

     ‘ncfinv’ uses Newton's method to converge to the solution.

     Further information about the noncentral F-distribution can be found at
     <https://en.wikipedia.org/wiki/Noncentral_F-distribution>

     See also: ncfcdf, ncfpdf, ncfrnd, ncfstat, finv.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 68
Inverse of the noncentral F-cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
ncfpdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 647
 -- statistics: Y = ncfpdf (X, DF1, DF2, LAMBDA)

     Noncentral F-probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the noncentral F-distribution with DF1 and DF2 degrees of freedom and
     noncentrality parameter LAMBDA.  The size of Y is the common size of X,
     DF1, DF2, and LAMBDA.  A scalar input functions as a constant matrix of the
     same size as the other inputs.

     Further information about the noncentral F-distribution can be found at
     <https://en.wikipedia.org/wiki/Noncentral_F-distribution>

     See also: ncfcdf, ncfinv, ncfrnd, ncfstat, fpdf.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 48
Noncentral F-probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
ncfrnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1368
 -- statistics: R = ncfrnd (DF1, DF2, LAMBDA)
 -- statistics: R = ncfrnd (DF1, DF2, LAMBDA, ROWS, COLS, ...)
 -- statistics: R = ncfrnd (DF1, DF2, LAMBDA, [SZ])

     Random arrays from the noncentral F-distribution.

     ‘X = ncfrnd (P, DF1, DF2, LAMBDA)’ returns an array of random numbers
     chosen from the noncentral F-distribution with DF1 and DF2 degrees of
     freedom and noncentrality parameter LAMBDA.  The size of R is the common
     size of DF1, DF2, and LAMBDA.  A scalar input functions as a constant
     matrix of the same size as the other input.

     ‘ncfrnd’ generates values using the definition of a noncentral F random
     variable, as the ratio of a noncentral chi-squared distribution and a
     (central) chi-squared distribution.

     When called with a single size argument, ‘ncfrnd’ returns a square matrix
     with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     Further information about the noncentral F-distribution can be found at
     <https://en.wikipedia.org/wiki/Noncentral_F-distribution>

     See also: ncfcdf, ncfinv, ncfpdf, ncfstat, frnd, ncx2rnd, chi2rnd.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 49
Random arrays from the noncentral F-distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
nctcdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 840
 -- statistics: P = nctcdf (X, DF, MU)
 -- statistics: P = nctcdf (X, DF, MU, "upper")

     Noncentral t-cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the noncentral t-distribution with DF degrees of freedom and
     noncentrality parameter MU.  The size of P is the common size of X, DF, and
     MU.  A scalar input functions as a constant matrix of the same size as the
     other inputs.

     ‘P = nctcdf (X, DF, MU, "upper")’ computes the upper tail probability of
     the noncentral t-distribution with parameters DF and MU, at the values in
     X.

     Further information about the noncentral t-distribution can be found at
     <https://en.wikipedia.org/wiki/Noncentral_t-distribution>

     See also: nctinv, nctpdf, nctrnd, nctstat, tcdf.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 52
Noncentral t-cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
nctinv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 699
 -- statistics: X = ncx2inv (P, DF, MU)

     Inverse of the non-central t-cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     noncentral t-distribution with DF degrees of freedom and noncentrality
     parameter MU.  The size of X is the common size of P, DF, and MU.  A scalar
     input functions as a constant matrix of the same size as the other inputs.

     ‘nctinv’ uses Newton's method to converge to the solution.

     Further information about the noncentral t-distribution can be found at
     <https://en.wikipedia.org/wiki/Noncentral_t-distribution>

     See also: nctcdf, nctpdf, nctrnd, nctstat, tinv.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 69
Inverse of the non-central t-cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
nctpdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 609
 -- statistics: Y = nctpdf (X, DF, MU)

     Noncentral t-probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the noncentral t-distribution with DF degrees of freedom and noncentrality
     parameter MU.  The size of Y is the common size of X, DF, and MU.  A scalar
     input functions as a constant matrix of the same size as the other inputs.

     Further information about the noncentral t-distribution can be found at
     <https://en.wikipedia.org/wiki/Noncentral_t-distribution>

     See also: nctcdf, nctinv, nctrnd, nctstat, tpdf.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 48
Noncentral t-probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
nctrnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1304
 -- statistics: R = nctrnd (DF, MU)
 -- statistics: R = nctrnd (DF, MU, ROWS, COLS, ...)
 -- statistics: R = nctrnd (DF, MU, [SZ])

     Random arrays from the noncentral t-distribution.

     ‘X = nctrnd (P, DF, MU)’ returns an array of random numbers chosen from the
     noncentral t-distribution with DF degrees of freedom and noncentrality
     parameter MU.  The size of R is the common size of DF and MU.  A scalar
     input functions as a constant matrix of the same size as the other input.

     ‘nctrnd’ generates values using the definition of a noncentral t random
     variable, as the ratio of a normal distribution with non-zero mean and the
     sqrt of a chi-squared distribution.

     When called with a single size argument, ‘nctrnd’ returns a square matrix
     with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     Further information about the noncentral t-distribution can be found at
     <https://en.wikipedia.org/wiki/Noncentral_t-distribution>

     See also: nctcdf, nctinv, nctpdf, nctstat, trnd, normrnd, chi2rnd.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 49
Random arrays from the noncentral t-distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
ncx2cdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 929
 -- statistics: P = ncx2cdf (X, DF, LAMBDA)
 -- statistics: P = ncx2cdf (X, DF, LAMBDA, "upper")

     Noncentral chi-squared cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the noncentral chi-squared distribution with DF degrees of freedom and
     noncentrality parameter LAMBDA.  The size of P is the common size of X, DF,
     and LAMBDA.  A scalar input functions as a constant matrix of the same size
     as the other inputs.

     ‘P = ncx2cdf (X, DF, LAMBDA, "upper")’ computes the upper tail probability
     of the noncentral chi-squared distribution with parameters DF and LAMBDA,
     at the values in X.

     Further information about the noncentral chi-squared distribution can be
     found at
     <https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution>

     See also: ncx2inv, ncx2pdf, ncx2rnd, ncx2stat, chi2cdf.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 62
Noncentral chi-squared cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
ncx2inv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 765
 -- statistics: X = ncx2inv (P, DF, LAMBDA)

     Inverse of the noncentral chi-squared cumulative distribution function
     (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     noncentral chi-squared distribution with DF degrees of freedom and
     noncentrality parameter MU.  The size of X is the common size of P, DF, and
     MU.  A scalar input functions as a constant matrix of the same size as the
     other inputs.

     ‘ncx2inv’ uses Newton's method to converge to the solution.

     Further information about the noncentral chi-squared distribution can be
     found at
     <https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution>

     See also: ncx2cdf, ncx2pdf, ncx2rnd, ncx2stat, chi2inv.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 78
Inverse of the noncentral chi-squared cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
ncx2pdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 684
 -- statistics: Y = ncx2pdf (X, DF, LAMBDA)

     Noncentral chi-squared probability distribution function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the noncentral chi-squared distribution with DF degrees of freedom and
     noncentrality parameter LAMBDA.  The size of Y is the common size of X, DF,
     and LAMBDA.  A scalar input functions as a constant matrix of the same size
     as the other inputs.

     Further information about the noncentral chi-squared distribution can be
     found at
     <https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution>

     See also: ncx2cdf, ncx2inv, ncx2rnd, ncx2stat, chi2pdf.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 63
Noncentral chi-squared probability distribution function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
ncx2rnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1157
 -- statistics: R = ncx2rnd (DF, LAMBDA)
 -- statistics: R = ncx2rnd (DF, LAMBDA, ROWS, COLS, ...)
 -- statistics: R = ncx2rnd (DF, LAMBDA, [SZ])

     Random arrays from the noncentral chi-squared distribution.

     ‘R = ncx2rnd (DF, LAMBDA)’ returns an array of random numbers chosen from
     the noncentral chi-squared distribution with DF degrees of freedom and
     noncentrality parameter LAMBDA.  The size of R is the common size of DF and
     LAMBDA.  A scalar input functions as a constant matrix of the same size as
     the other input.

     When called with a single size argument, ‘ncx2rnd’ returns a square matrix
     with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     Further information about the noncentral chi-squared distribution can be
     found at
     <https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution>

     See also: ncx2cdf, ncx2inv, ncx2pdf, ncx2stat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 59
Random arrays from the noncentral chi-squared distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
normcdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1765
 -- statistics: P = normcdf (X)
 -- statistics: P = normcdf (X, MU)
 -- statistics: P = normcdf (X, MU, SIGMA)
 -- statistics: P = normcdf (..., "upper")
 -- statistics: [P, PLO, PUP] = normcdf (X, MU, SIGMA, PCOV)
 -- statistics: [P, PLO, PUP] = normcdf (X, MU, SIGMA, PCOV, ALPHA)
 -- statistics: [P, PLO, PUP] = normcdf (..., "upper")

     Normal cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the normal distribution with mean MU and standard deviation SIGMA.  The
     size of P is the common size of X, MU and SIGMA.  A scalar input functions
     as a constant matrix of the same size as the other inputs.

     Default values are MU = 0, SIGMA = 1.

     When called with three output arguments, i.e.  [P, PLO, PUP], ‘normcdf’
     computes the confidence bounds for P when the input parameters MU and SIGMA
     are estimates.  In such case, PCOV, a 2x2 matrix containing the covariance
     matrix of the estimated parameters, is necessary.  Optionally, ALPHA, which
     has a default value of 0.05, specifies the 100 * (1 - ALPHA) percent
     confidence bounds.  PLO and PUP are arrays of the same size as P containing
     the lower and upper confidence bounds.

     ‘[...] = normcdf (..., "upper")’ computes the upper tail probability of the
     normal distribution with parameters MU and SIGMA, at the values in X.  This
     can be used to compute a right-tailed p-value.  To compute a two-tailed
     p-value, use ‘2 * normcdf (-abs (X), MU, SIGMA)’.

     Further information about the normal distribution can be found at
     <https://en.wikipedia.org/wiki/Normal_distribution>

     See also: norminv, normpdf, normrnd, normfit, normlike, normstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 46
Normal cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
norminv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 960
 -- statistics: X = norminv (P)
 -- statistics: X = norminv (P, MU)
 -- statistics: X = norminv (P, MU, SIGMA)

     Inverse of the normal cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     normal distribution with mean MU and standard deviation SIGMA.  The size of
     P is the common size of P, MU and SIGMA.  A scalar input functions as a
     constant matrix of the same size as the other inputs.

     Default values are MU = 0, SIGMA = 1.

     The default values correspond to the standard normal distribution and
     computing its quantile function is also possible with the ‘probit’
     function, which is faster but it does not perform any input validation.

     Further information about the normal distribution can be found at
     <https://en.wikipedia.org/wiki/Normal_distribution>

     See also: norminv, normpdf, normrnd, normfit, normlike, normstat, probit.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 62
Inverse of the normal cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
normpdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 704
 -- statistics: Y = normpdf (X)
 -- statistics: Y = normpdf (X, MU)
 -- statistics: Y = normpdf (X, MU, SIGMA)

     Normal probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the normal distribution with mean MU and standard deviation SIGMA.  The
     size of Y is the common size of P, MU and SIGMA.  A scalar input functions
     as a constant matrix of the same size as the other inputs.

     Default values are MU = 0, SIGMA = 1.

     Further information about the normal distribution can be found at
     <https://en.wikipedia.org/wiki/Normal_distribution>

     See also: norminv, norminv, normrnd, normfit, normlike, normstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 42
Normal probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
normrnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1216
 -- statistics: R = normrnd (MU, SIGMA)
 -- statistics: R = normrnd (MU, SIGMA, ROWS)
 -- statistics: R = normrnd (MU, SIGMA, ROWS, COLS, ...)
 -- statistics: R = normrnd (MU, SIGMA, [SZ])

     Random arrays from the normal distribution.

     ‘R = normrnd (MU, SIGMA)’ returns an array of random numbers chosen from
     the normal distribution with mean MU and standard deviation SIGMA.  The
     size of R is the common size of MU and SIGMA.  A scalar input functions as
     a constant matrix of the same size as the other inputs.  Both parameters
     must be finite real numbers and SIGMA > 0, otherwise NaN is returned.

     When called with a single size argument, ‘normrnd’ returns a square matrix
     with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     Further information about the normal distribution can be found at
     <https://en.wikipedia.org/wiki/Normal_distribution>

     See also: norminv, norminv, normpdf, normfit, normlike, normstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 43
Random arrays from the normal distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
plcdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 895
 -- statistics: P = plcdf (DATA, X, FX)
 -- statistics: P = plcdf (DATA, X, FX, "upper")

     Piecewise linear cumulative distribution function (CDF).

     For each element of DATA, compute the cumulative distribution function
     (CDF) of the piecewise linear distribution with a vector of X values at
     which the CDF changes slope and a vector of CDF values FX that correspond
     to each value in X.  Both X and FX must be vectors of the same size and at
     least 2-elements long.  The size of P is the same as DATA.

     ‘P = plcdf (DATA, X, FX, "upper")’ computes the upper tail probability of
     the piecewise linear distribution with parameters X and FX, at the values
     in DATA.

     Further information about the piecewise linear distribution can be found at
     <https://en.wikipedia.org/wiki/Piecewise_linear_function>

     See also: plinv, plpdf, plrnd, plstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 56
Piecewise linear cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
plinv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 662
 -- statistics: DATA = plinv (P, X, FX)

     Inverse of the piecewise linear distribution (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     piecewise linear distribution with a vector of X values at which the CDF
     changes slope and a vector of CDF values FX that correspond to each value
     in X.  Both X and FX must be vectors of the same_p size and at least
     2-elements long..  The size of DATA is the same_p as P.

     Further information about the piecewise linear distribution can be found at
     <https://en.wikipedia.org/wiki/Piecewise_linear_function>

     See also: plcdf, plpdf, plrnd, plstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 52
Inverse of the piecewise linear distribution (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
plpdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 661
 -- statistics: Y = plpdf (DATA, X, FX)

     Piecewise linear probability density function (PDF).

     For each element of DATA, compute the probability density function (PDF) of
     the piecewise linear distribution with a vector of X values at which the
     CDF changes slope and a vector of CDF values FX that correspond to each
     value in X.  Both X and FX must be vectors of the same size and at least
     2-elements long.  The size of P is the same as DATA.

     Further information about the piecewise linear distribution can be found at
     <https://en.wikipedia.org/wiki/Piecewise_linear_function>

     See also: plcdf, plinv, plrnd, plstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 52
Piecewise linear probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
plrnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1116
 -- statistics: R = plrnd (X, FX)
 -- statistics: R = plrnd (X, FX, ROWS)
 -- statistics: R = plrnd (X, FX, ROWS, COLS, ...)
 -- statistics: R = plrnd (X, FX, [SZ])

     Random arrays from the piecewise linear distribution.

     ‘R = plrnd (X, FX)’ returns a random number chosen from the piecewise
     linear distribution with a vector of X values at which the CDF changes
     slope and a vector of CDF values FX that correspond to each value in X.
     Both X and FX must be vectors of the same size and at least 2-elements
     long.

     When called with a single size argument, ‘plrnd’ returns a square matrix
     with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     Further information about the piecewise linear distribution can be found at
     <https://en.wikipedia.org/wiki/Piecewise_linear_function>

     See also: plcdf, plinv, plpdf, plstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 53
Random arrays from the piecewise linear distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
poisscdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 797
 -- statistics: P = poisscdf (X, LAMBDA)
 -- statistics: P = poisscdf (X, LAMBDA, "upper")

     Poisson cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the Poisson distribution with rate parameter LAMBDA.  The size of P is
     the common size of X and LAMBDA.  A scalar input functions as a constant
     matrix of the same size as the other inputs.

     ‘P = poisscdf (X, LAMBDA, "upper")’ computes the upper tail probability of
     the Poisson distribution with parameter LAMBDA, at the values in X.

     Further information about the Poisson distribution can be found at
     <https://en.wikipedia.org/wiki/Poisson_distribution>

     See also: poissinv, poisspdf, poissrnd, poissfit, poisslike, poisstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 47
Poisson cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
poissinv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 600
 -- statistics: X = poissinv (P, LAMBDA)

     Inverse of the Poisson cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     Poisson distribution with rate parameter LAMBDA.  The size of X is the
     common size of P and LAMBDA.  A scalar input functions as a constant matrix
     of the same size as the other inputs.

     Further information about the Poisson distribution can be found at
     <https://en.wikipedia.org/wiki/Poisson_distribution>

     See also: poisscdf, poisspdf, poissrnd, poissfit, poisslike, poisstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 63
Inverse of the Poisson cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
poisspdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 581
 -- statistics: Y = poisspdf (X, LAMBDA)

     Poisson probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the Poisson distribution with rate parameter LAMBDA.  The size of Y is the
     common size of X and LAMBDA.  A scalar input functions as a constant matrix
     of the same size as the other inputs.

     Further information about the Poisson distribution can be found at
     <https://en.wikipedia.org/wiki/Poisson_distribution>

     See also: poisscdf, poissinv, poissrnd, poissfit, poisslike, poisstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 43
Poisson probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
poissrnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1198
 -- statistics: R = poissrnd (LAMBDA)
 -- statistics: R = poissrnd (LAMBDA, ROWS)
 -- statistics: R = poissrnd (LAMBDA, ROWS, COLS, ...)
 -- statistics: R = poissrnd (LAMBDA, [SZ])

     Random arrays from the Poisson distribution.

     ‘R = normrnd (LAMBDA)’ returns an array of random numbers chosen from the
     Poisson distribution with rate parameter LAMBDA.  The size of R is the
     common size of LAMBDA.  A scalar input functions as a constant matrix of
     the same size as the other inputs.  LAMBDA must be a finite real number and
     greater or equal to 0, otherwise NaN is returned.

     When called with a single size argument, ‘poissrnd’ returns a square matrix
     with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     Further information about the Poisson distribution can be found at
     <https://en.wikipedia.org/wiki/Poisson_distribution>

     See also: poisscdf, poissinv, poisspdf, poissfit, poisslike, poisstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 44
Random arrays from the Poisson distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
raylcdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 789
 -- statistics: P = raylcdf (X, SIGMA)
 -- statistics: P = raylcdf (X, SIGMA, "upper")

     Rayleigh cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the Rayleigh distribution with scale parameter SIGMA.  The size of P is
     the common size of X and SIGMA.  A scalar input functions as a constant
     matrix of the same size as the other inputs.

     ‘P = raylcdf (X, SIGMA, "upper")’ computes the upper tail probability of
     the Rayleigh distribution with parameter SIGMA, at the values in X.

     Further information about the Rayleigh distribution can be found at
     <https://en.wikipedia.org/wiki/Rayleigh_distribution>

     See also: raylinv, raylpdf, raylrnd, raylfit, rayllike, raylstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 48
Rayleigh cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
raylinv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 596
 -- statistics: X = raylinv (P, SIGMA)

     Inverse of the Rayleigh cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     Rayleigh distribution with scale parameter SIGMA.  The size of X is the
     common size of P and SIGMA.  A scalar input functions as a constant matrix
     of the same size as the other inputs.

     Further information about the Rayleigh distribution can be found at
     <https://en.wikipedia.org/wiki/Rayleigh_distribution>

     See also: raylcdf, raylpdf, raylrnd, raylfit, rayllike, raylstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 64
Inverse of the Rayleigh cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
raylpdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 577
 -- statistics: Y = raylpdf (X, SIGMA)

     Rayleigh probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the Rayleigh distribution with scale parameter SIGMA.  The size of P is the
     common size of X and SIGMA.  A scalar input functions as a constant matrix
     of the same size as the other inputs.

     Further information about the Rayleigh distribution can be found at
     <https://en.wikipedia.org/wiki/Rayleigh_distribution>

     See also: raylcdf, raylinv, raylrnd, raylfit, rayllike, raylstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 44
Rayleigh probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
raylrnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1167
 -- statistics: R = raylrnd (SIGMA)
 -- statistics: R = raylrnd (SIGMA, ROWS)
 -- statistics: R = raylrnd (SIGMA, ROWS, COLS, ...)
 -- statistics: R = raylrnd (SIGMA, [SZ])

     Random arrays from the Rayleigh distribution.

     ‘R = raylrnd (SIGMA)’ returns an array of random numbers chosen from the
     Rayleigh distribution with scale parameter SIGMA.  The size of R is the
     size of SIGMA.  A scalar input functions as a constant matrix of the same
     size as the other inputs.  SIGMA must be a finite real number greater than
     0, otherwise NaN is returned.

     When called with a single size argument, ‘raylrnd’ returns a square matrix
     with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     Further information about the Rayleigh distribution can be found at
     <https://en.wikipedia.org/wiki/Rayleigh_distribution>

     See also: raylcdf, raylinv, raylpdf, raylfit, rayllike, raylstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 45
Random arrays from the Rayleigh distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
ricecdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 844
 -- statistics: P = ricecdf (X, S, SIGMA)
 -- statistics: P = ricecdf (X, S, SIGMA, "upper")

     Rician cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the Rician distribution with non-centrality (distance) parameter S and
     scale parameter SIGMA.  The size of P is the common size of X, S, and
     SIGMA.  A scalar input functions as a constant matrix of the same size as
     the other inputs.

     ‘P = ricecdf (X, S, SIGMA, "upper")’ computes the upper tail probability of
     the Rician distribution with parameters S and SIGMA, at the values in X.

     Further information about the Rician distribution can be found at
     <https://en.wikipedia.org/wiki/Rice_distribution>

     See also: riceinv, ricepdf, ricernd, ricefit, ricelike, ricestat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 46
Rician cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
riceinv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 620
 -- statistics: X = riceinv (P, S, SIGMA)

     Inverse of the Rician distribution (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     Rician distribution with non-centrality (distance) parameter S and scale
     parameter SIGMA.  The size of X is the common size of X, S, and SIGMA.  A
     scalar input functions as a constant matrix of the same size as the other
     inputs.

     Further information about the Rician distribution can be found at
     <https://en.wikipedia.org/wiki/Rice_distribution>

     See also: ricecdf, ricepdf, ricernd, ricefit, ricelike, ricestat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 42
Inverse of the Rician distribution (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
ricepdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 621
 -- statistics: Y = ricepdf (X, S, SIGMA)

     Rician probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the Rician distribution with non-centrality (distance) parameter S and
     scale parameter SIGMA.  The size of Y is the common size of X, S, and
     SIGMA.  A scalar input functions as a constant matrix of the same size as
     the other inputs.

     Further information about the Rician distribution can be found at
     <https://en.wikipedia.org/wiki/Rice_distribution>

     See also: ricecdf, riceinv, ricernd, ricefit, ricelike, ricestat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 42
Rician probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
ricernd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1131
 -- statistics: R = ricernd (S, SIGMA)
 -- statistics: R = ricernd (S, SIGMA, ROWS)
 -- statistics: R = ricernd (S, SIGMA, ROWS, COLS, ...)
 -- statistics: R = ricernd (S, SIGMA, [SZ])

     Random arrays from the Rician distribution.

     ‘R = ricernd (S, SIGMA)’ returns an array of random numbers chosen from the
     Rician distribution with noncentrality parameter S and scale parameter
     SIGMA.  The size of R is the common size of S and SIGMA.  A scalar input
     functions as a constant matrix of the same size as the other inputs.

     When called with a single size argument, ‘ricernd’ returns a square matrix
     with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     Further information about the Rician distribution can be found at
     <https://en.wikipedia.org/wiki/Rice_distribution>

     See also: ricecdf, riceinv, ricepdf, ricefit, ricelike, ricestat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 43
Random arrays from the Rician distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 4
tcdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 759
 -- statistics: P = tcdf (X, DF)
 -- statistics: P = tcdf (X, DF, "upper")

     Student's T cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the Student's T distribution with DF degrees of freedom.  The size of P
     is the common size of X and DF.  A scalar input functions as a constant
     matrix of the same size as the other input.

     ‘P = tcdf (X, DF, "upper")’ computes the upper tail probability of the
     Student's T distribution with DF degrees of freedom, at the values in X.

     Further information about the Student's T distribution can be found at
     <https://en.wikipedia.org/wiki/Student%27s_t-distribution>

     See also: tinv, tpdf, trnd, tstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 51
Student's T cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 4
tinv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 769
 -- statistics: X = tinv (P, DF)

     Inverse of the Student's T cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     Student's T distribution with DF degrees of freedom.  The size of X is the
     common size of X and DF.  A scalar input functions as a constant matrix of
     the same size as the other input.

     This function is analogous to looking in a table for the t-value of a
     single-tailed distribution.  For very large DF (>10000), the inverse of the
     standard normal distribution is used.

     Further information about the Student's T distribution can be found at
     <https://en.wikipedia.org/wiki/Student%27s_t-distribution>

     See also: tcdf, tpdf, trnd, tstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 67
Inverse of the Student's T cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
tlscdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 998
 -- statistics: P = tlscdf (X, MU, SIGMA, NU)
 -- statistics: P = tlscdf (X, MU, SIGMA, NU, "upper")

     Location-scale Student's T cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the location-scale Student's T distribution with location parameter MU,
     scale parameter SIGMA, and NU degrees of freedom.  The size of P is the
     common size of X, MU, SIGMA, and NU.  A scalar input functions as a
     constant matrix of the same size as the other inputs.

     ‘P = tlscdf (X, MU, SIGMA, NU, "upper")’ computes the upper tail
     probability of the location-scale Student's T distribution with parameters
     MU, SIGMA, and NU, at the values in X.

     Further information about the location-scale Student's T distribution can
     be found at
     <https://en.wikipedia.org/wiki/Student%27s_t-distribution#Location-scale_t_distribution>

     See also: tlsinv, tlspdf, tlsrnd, tlsfit, tlslike, tlsstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 66
Location-scale Student's T cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
tlsinv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 760
 -- statistics: X = tlsinv (P, MU, SIGMA, NU)

     Inverse of the location-scale Student's T cumulative distribution function
     (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     location-scale Student's T distribution with location parameter MU, scale
     parameter SIGMA, and NU degrees of freedom.  The size of X is the common
     size of P, MU, SIGMA, and NU.  A scalar input functions as a constant
     matrix of the same size as the other inputs.

     Further information about the location-scale Student's T distribution can
     be found at
     <https://en.wikipedia.org/wiki/Student%27s_t-distribution#Location-scale_t_distribution>

     See also: tlscdf, tlspdf, tlsrnd, tlsfit, tlslike, tlsstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Inverse of the location-scale Student's T cumulative distribution function
(i...



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
tlspdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 736
 -- statistics: P = tlspdf (X, MU, SIGMA, NU)

     Location-scale Student's T probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the location-scale Student's T distribution with location parameter MU,
     scale parameter SIGMA, and NU degrees of freedom.  The size of Y is the
     common size of X, MU, SIGMA, and NU.  A scalar input functions as a
     constant matrix of the same size as the other inputs.

     Further information about the location-scale Student's T distribution can
     be found at
     <https://en.wikipedia.org/wiki/Student%27s_t-distribution#Location-scale_t_distribution>

     See also: tlscdf, tlsinv, tlsrnd, tlsfit, tlslike, tlsstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 62
Location-scale Student's T probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
tlsrnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1445
 -- statistics: R = tlsrnd (MU, SIGMA, NU)
 -- statistics: R = tlsrnd (MU, SIGMA, NU, ROWS)
 -- statistics: R = tlsrnd (MU, SIGMA, NU, ROWS, COLS, ...)
 -- statistics: R = tlsrnd (MU, SIGMA, NU, [SZ])

     Random arrays from the location-scale Student's T distribution.

     Return a matrix of random samples from the location-scale Student's T
     distribution with location parameter MU, scale parameter SIGMA, and NU
     degrees of freedom.

     ‘R = tlsrnd (NU)’ returns an array of random numbers chosen from the
     location-scale Student's T distribution with location parameter MU, scale
     parameter SIGMA, and NU degrees of freedom.  The size of R is the common
     size of MU, SIGMA, and NU.  A scalar input functions as a constant matrix
     of the same size as the other inputs.

     When called with a single size argument, ‘tlsrnd’ returns a square matrix
     with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     Further information about the location-scale Student's T distribution can
     be found at
     <https://en.wikipedia.org/wiki/Student%27s_t-distribution#Location-scale_t_distribution>

     See also: tlscdf, tlsinv, tlspdf, tlsfit, tlslike, tlsstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 63
Random arrays from the location-scale Student's T distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 4
tpdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 550
 -- statistics: P = tpdf (X, DF)

     Student's T probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the Student's T distribution with DF degrees of freedom.  The size of Y is
     the common size of X and DF.  A scalar input functions as a constant matrix
     of the same size as the other input.

     Further information about the Student's T distribution can be found at
     <https://en.wikipedia.org/wiki/Student%27s_t-distribution>

     See also: tcdf, tpdf, trnd, tstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 47
Student's T probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
tricdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1369
 -- statistics: P = tricdf (X, A, B, C)
 -- statistics: P = tricdf (X, A, B, C, "upper")

     Triangular cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the triangular distribution with lower limit parameter A, peak location
     (mode) parameter B, and upper limit parameter C.  The size of P is the
     common size of the input arguments.  A scalar input functions as a constant
     matrix of the same size as the other inputs.

     ‘P = tricdf (X, A, B, C, "upper")’ computes the upper tail probability of
     the triangular distribution with parameters A, B, and C, at the values in
     X.

     Note that the order of the parameter input arguments has been changed after
     statistics version 1.6.3 in order to be MATLAB compatible with the
     parameters used in the TriangularDistribution probability distribution
     object.  More specifically, the positions of the parameters B and C have
     been swapped.  As a result, the naming conventions no longer coincide with
     those used in Wikipedia, in which b denotes the upper limit and c denotes
     the mode or peak parameter.

     Further information about the triangular distribution can be found at
     <https://en.wikipedia.org/wiki/Triangular_distribution>

     See also: triinv, tripdf, trirnd, tristat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 50
Triangular cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
triinv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1160
 -- statistics: X = triinv (P, A, B, C)

     Inverse of the triangular cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     triangular distribution with lower limit parameter A, peak location (mode)
     parameter B, and upper limit parameter C.  The size of X is the common size
     of the input arguments.  A scalar input functions as a constant matrix of
     the same size as the other inputs.

     Note that the order of the parameter input arguments has been changed after
     statistics version 1.6.3 in order to be MATLAB compatible with the
     parameters used in the TriangularDistribution probability distribution
     object.  More specifically, the positions of the parameters B and C have
     been swapped.  As a result, the naming conventions no longer coincide with
     those used in Wikipedia, in which b denotes the upper limit and c denotes
     the mode or peak parameter.

     Further information about the triangular distribution can be found at
     <https://en.wikipedia.org/wiki/Triangular_distribution>

     See also: tricdf, tripdf, trirnd, tristat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 66
Inverse of the triangular cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
tripdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1141
 -- statistics: Y = tripdf (X, A, B, C)

     Triangular probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the triangular distribution with lower limit parameter A, peak location
     (mode) parameter B, and upper limit parameter C.  The size of Y is the
     common size of the input arguments.  A scalar input functions as a constant
     matrix of the same size as the other inputs.

     Note that the order of the parameter input arguments has been changed after
     statistics version 1.6.3 in order to be MATLAB compatible with the
     parameters used in the TriangularDistribution probability distribution
     object.  More specifically, the positions of the parameters B and C have
     been swapped.  As a result, the naming conventions no longer coincide with
     those used in Wikipedia, in which b denotes the upper limit and c denotes
     the mode or peak parameter.

     Further information about the triangular distribution can be found at
     <https://en.wikipedia.org/wiki/Triangular_distribution>

     See also: tricdf, triinv, trirnd, tristat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 46
Triangular probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
trirnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1653
 -- statistics: R = trirnd (A, B, C)
 -- statistics: R = trirnd (A, B, C, ROWS)
 -- statistics: R = trirnd (A, B, C, ROWS, COLS, ...)
 -- statistics: R = trirnd (A, B, C, [SZ])

     Random arrays from the triangular distribution.

     ‘R = trirnd (SIGMA)’ returns an array of random numbers chosen from the
     triangular distribution with lower limit parameter A, peak location (mode)
     parameter B, and upper limit parameter C.  The size of R is the common size
     of A, B, and C.  A scalar input functions as a constant matrix of the same
     size as the other inputs.

     When called with a single size argument, ‘trirnd’ returns a square matrix
     with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     Note that the order of the parameter input arguments has been changed after
     statistics version 1.6.3 in order to be MATLAB compatible with the
     parameters used in the TriangularDistribution probability distribution
     object.  More specifically, the positions of the parameters B and C have
     been swapped.  As a result, the naming conventions no longer coincide with
     those used in Wikipedia, in which b denotes the upper limit and c denotes
     the mode or peak parameter.

     Further information about the triangular distribution can be found at
     <https://en.wikipedia.org/wiki/Triangular_distribution>

     See also: tricdf, triinv, tripdf, tristat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 47
Random arrays from the triangular distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 4
trnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1219
 -- statistics: R = trnd (DF)
 -- statistics: R = trnd (DF, ROWS)
 -- statistics: R = trnd (DF, ROWS, COLS, ...)
 -- statistics: R = trnd (DF, [SZ])

     Random arrays from the Student's T distribution.

     Return a matrix of random samples from the Students's T distribution with
     DF degrees of freedom.

     ‘R = trnd (DF)’ returns an array of random numbers chosen from the
     Student's T distribution with DF degrees of freedom.  The size of R is the
     size of DF.  A scalar input functions as a constant matrix of the same size
     as the other inputs.  DF must be a finite real number greater than 0,
     otherwise NaN is returned.

     When called with a single size argument, ‘trnd’ returns a square matrix
     with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     Further information about the Student's T distribution can be found at
     <https://en.wikipedia.org/wiki/Student%27s_t-distribution>

     See also: tcdf, tpdf, tpdf, tstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 48
Random arrays from the Student's T distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
unidcdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1219
 -- statistics: P = unidcdf (X, N)
 -- statistics: P = unidcdf (X, N, "upper")

     Discrete uniform cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of a discrete uniform distribution with parameter N, which corresponds to
     the maximum observable value.  ‘unidcdf’ assumes the integer values in the
     range [1,N] with equal probability.  The size of P is the common size of X
     and N.  A scalar input functions as a constant matrix of the same size as
     the other inputs.

     The maximum observable values in N must be positive integers, otherwise NaN
     is returned.

     ‘[...] = unidcdf (X, N, "upper")’ computes the upper tail probability of
     the discrete uniform distribution with maximum observable value N, at the
     values in X.

     Warning: The underlying implementation uses the double class and will only
     be accurate for N < ‘flintmax’ (2^{53} on IEEE 754 compatible systems).

     Further information about the discrete uniform distribution can be found at
     <https://en.wikipedia.org/wiki/Discrete_uniform_distribution>

     See also: unidinv, unidpdf, unidrnd, unidfit, unidstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 56
Discrete uniform cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
unidinv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1008
 -- statistics: X = unidinv (P, N)

     Inverse of the discrete uniform cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     discrete uniform distribution with parameter N, which corresponds to the
     maximum observable value.  ‘unidinv’ assumes the integer values in the
     range [1,N] with equal probability.  The size of X is the common size of P
     and N.  A scalar input functions as a constant matrix of the same size as
     the other inputs.

     The maximum observable values in N must be positive integers, otherwise NaN
     is returned.

     Warning: The underlying implementation uses the double class and will only
     be accurate for N < ‘flintmax’ (2^{53} on IEEE 754 compatible systems).

     Further information about the discrete uniform distribution can be found at
     <https://en.wikipedia.org/wiki/Discrete_uniform_distribution>

     See also: unidcdf, unidpdf, unidrnd, unidfit, unidstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 72
Inverse of the discrete uniform cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
unidpdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 989
 -- statistics: Y = unidpdf (X, N)

     Discrete uniform probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the discrete uniform distribution with parameter N, which corresponds to
     the maximum observable value.  ‘unidpdf’ assumes the integer values in the
     range [1,N] with equal probability.  The size of X is the common size of P
     and N.  A scalar input functions as a constant matrix of the same size as
     the other inputs.

     The maximum observable values in N must be positive integers, otherwise NaN
     is returned.

     Warning: The underlying implementation uses the double class and will only
     be accurate for N < ‘flintmax’ (2^{53} on IEEE 754 compatible systems).

     Further information about the discrete uniform distribution can be found at
     <https://en.wikipedia.org/wiki/Discrete_uniform_distribution>

     See also: unidcdf, unidinv, unidrnd, unidfit, unidstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 52
Discrete uniform probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
unidrnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1479
 -- statistics: R = unidrnd (N)
 -- statistics: R = unidrnd (N, ROWS)
 -- statistics: R = unidrnd (N, ROWS, COLS, ...)
 -- statistics: R = unidrnd (N, [SZ])

     Random arrays from the discrete uniform distribution.

     ‘R = unidrnd (N)’ returns an array of random numbers chosen from the
     discrete uniform distribution with parameter N, which corresponds to the
     maximum observable value.  ‘unidrnd’ assumes the integer values in the
     range [1,N] with equal probability.  The size of R is the size of N.  A
     scalar input functions as a constant matrix of the same size as the other
     inputs.

     The maximum observable values in N must be positive integers, otherwise NaN
     is returned.

     When called with a single size argument, ‘unidrnd’ returns a square matrix
     with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     Warning: The underlying implementation uses the double class and will only
     be accurate for N < ‘flintmax’ (2^{53} on IEEE 754 compatible systems).

     Further information about the discrete uniform distribution can be found at
     <https://en.wikipedia.org/wiki/Discrete_uniform_distribution>

     See also: unidcdf, unidinv, unidrnd, unidfit, unidstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 53
Random arrays from the discrete uniform distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
unifcdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 904
 -- statistics: P = unifcdf (X, A, B)
 -- statistics: P = unifcdf (X, A, B, "upper")

     Continuous uniform cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the continuous uniform distribution with parameters A and B, which
     define the lower and upper bounds of the interval [A, B].  The size of P is
     the common size of X, A, and B.  A scalar input functions as a constant
     matrix of the same size as the other inputs.

     ‘[...] = unifcdf (X, A, B, "upper")’ computes the upper tail probability of
     the continuous uniform distribution with parameters A, and B, at the values
     in X.

     Further information about the continuous uniform distribution can be found
     at <https://en.wikipedia.org/wiki/Continuous_uniform_distribution>

     See also: unifinv, unifpdf, unifrnd, unifit, unifstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 58
Continuous uniform cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
unifinv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 690
 -- statistics: X = unifinv (P, A, B)

     Inverse of the continuous uniform cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     continuous uniform distribution with parameters A and B, which define the
     lower and upper bounds of the interval [A, B].  The size of X is the common
     size of P, A, and B.  A scalar input functions as a constant matrix of the
     same size as the other inputs.

     Further information about the continuous uniform distribution can be found
     at <https://en.wikipedia.org/wiki/Continuous_uniform_distribution>

     See also: unifcdf, unifpdf, unifrnd, unifit, unifstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 74
Inverse of the continuous uniform cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
unifpdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 671
 -- statistics: Y = unifpdf (X, A, B)

     Continuous uniform probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the continuous uniform distribution with parameters A and B, which define
     the lower and upper bounds of the interval [A, B].  The size of Y is the
     common size of X, A, and B.  A scalar input functions as a constant matrix
     of the same size as the other inputs.

     Further information about the continuous uniform distribution can be found
     at <https://en.wikipedia.org/wiki/Continuous_uniform_distribution>

     See also: unifcdf, unifinv, unifrnd, unifit, unifstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 54
Continuous uniform probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
unifrnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1182
 -- statistics: R = unifrnd (A, B)
 -- statistics: R = unifrnd (A, B, ROWS)
 -- statistics: R = unifrnd (A, B, ROWS, COLS, ...)
 -- statistics: R = unifrnd (A, B, [SZ])

     Random arrays from the continuous uniform distribution.

     ‘R = unifrnd (A, B)’ returns an array of random numbers chosen from the
     continuous uniform distribution with parameters A and B, which define the
     lower and upper bounds of the interval [A, B].  The size of R is the common
     size of A and B.  A scalar input functions as a constant matrix of the same
     size as the other inputs.

     When called with a single size argument, ‘unifrnd’ returns a square matrix
     with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     Further information about the continuous uniform distribution can be found
     at <https://en.wikipedia.org/wiki/Continuous_uniform_distribution>

     See also: unifcdf, unifinv, unifpdf, unifit, unifstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 55
Random arrays from the continuous uniform distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
vmcdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1190
 -- statistics: P = vmcdf (X, MU, K)
 -- statistics: P = vmcdf (X, MU, K, "upper")

     Von Mises probability density function (PDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the von Mises distribution with location parameter MU and concentration
     parameter K on the interval [-pi,pi].  The size of P is the common size of
     X, MU, and K.  A scalar input functions as a constant matrix of the same
     same size as the other inputs.

     ‘P = vmcdf (X, MU, K, "upper")’ computes the upper tail probability of the
     von Mises distribution with parameters MU and K, at the values in X.

     Note: the CDF of the von Mises distribution is not analytic.  Hence, it is
     calculated by integrating its probability density which is expressed as a
     series of Bessel functions.  Balancing between performance and accuracy,
     the integration uses a step of 1e-5 on the interval [-pi,pi], which results
     to an accuracy of about 10 significant digits.

     Further information about the von Mises distribution can be found at
     <https://en.wikipedia.org/wiki/Von_Mises_distribution>

     See also: vminv, vmpdf, vmrnd.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 45
Von Mises probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
vminv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 985
 -- statistics: X = vminv (P, MU, K)

     Inverse of the von Mises cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     von Mises distribution with location parameter MU and concentration
     parameter K on the interval [-pi,pi].  The size of X is the common size of
     P, MU, and K.  A scalar input functions as a constant matrix of the same
     size as the other inputs.

     Note: the quantile of the von Mises distribution is not analytic.  Hence,
     it is approximated by a custom searching algorithm using its CDF until it
     converges up to a tolerance of 1e-5 or 100 iterations.  As a result,
     balancing between performance and accuracy, the accuracy is about 5e-5 for
     K = 1 and it drops to 5e-5 as K increases.

     Further information about the von Mises distribution can be found at
     <https://en.wikipedia.org/wiki/Von_Mises_distribution>

     See also: vmcdf, vmpdf, vmrnd.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 65
Inverse of the von Mises cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
vmpdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 606
 -- statistics: Y = vmpdf (X, MU, K)

     Von Mises probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the von Mises distribution with location parameter MU and concentration
     parameter K on the interval [-pi, pi].  The size of Y is the common size of
     X, MU, and K.  A scalar input functions as a constant matrix of the same
     size as the other inputs.

     Further information about the von Mises distribution can be found at
     <https://en.wikipedia.org/wiki/Von_Mises_distribution>

     See also: vmcdf, vminv, vmrnd.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 45
Von Mises probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
vmrnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1195
 -- statistics: R = vmrnd (MU, K)
 -- statistics: R = vmrnd (MU, K, ROWS)
 -- statistics: R = vmrnd (MU, K, ROWS, COLS, ...)
 -- statistics: R = vmrnd (MU, K, [SZ])

     Random arrays from the von Mises distribution.

     ‘R = vmrnd (MU, K)’ returns an array of random angles chosen from a von
     Mises distribution with location parameter MU and concentration parameter K
     on the interval [-pi, pi].  The size of R is the common size of MU and K.
     A scalar input functions as a constant matrix of the same size as the other
     inputs.  Both parameters must be finite real numbers and K > 0, otherwise
     NaN is returned.

     When called with a single size argument, ‘vmrnd’ returns a square matrix
     with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     Further information about the von Mises distribution can be found at
     <https://en.wikipedia.org/wiki/Von_Mises_distribution>

     See also: vmcdf, vminv, vmpdf.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 46
Random arrays from the von Mises distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
wblcdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1547
 -- statistics: P = wblcdf (X)
 -- statistics: P = wblcdf (X, LAMBDA)
 -- statistics: P = wblcdf (X, LAMBDA, K)
 -- statistics: P = wblcdf (..., "upper")
 -- statistics: [P, PLO, PUP] = wblcdf (X, LAMBDA, K, PCOV)
 -- statistics: [P, PLO, PUP] = wblcdf (X, LAMBDA, K, PCOV, ALPHA)
 -- statistics: [P, PLO, PUP] = wblcdf (..., "upper")

     Weibull cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function (CDF)
     of the Weibull distribution with scale parameter LAMBDA and shape parameter
     K.  The size of P is the common size of X, LAMBDA and K.  A scalar input
     functions as a constant matrix of the same size as the other inputs.

     Default values are LAMBDA = 1, K = 1.

     When called with three output arguments, ‘[P, PLO, PUP]’ it computes the
     confidence bounds for P when the input parameters LAMBDA and K are
     estimates.  In such case, PCOV, a 2-by-2 matrix containing the covariance
     matrix of the estimated parameters, is necessary.  Optionally, ALPHA has a
     default value of 0.05, and specifies 100 * (1 - ALPHA)% confidence bounds.
     PLO and PUP are arrays of the same size as P containing the lower and upper
     confidence bounds.

     ‘[...] = wblcdf (..., "upper")’ computes the upper tail probability of the
     lognormal distribution.

     Further information about the Weibull distribution can be found at
     <https://en.wikipedia.org/wiki/Weibull_distribution>

     See also: wblinv, wblpdf, wblrnd, wblstat, wblplot.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 47
Weibull cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
wblinv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 722
 -- statistics: X = wblinv (P)
 -- statistics: X = wblinv (P, LAMBDA)
 -- statistics: X = wblinv (P, LAMBDA, K)

     Inverse of the Weibull cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the CDF) of the
     Weibull distribution with scale parameter LAMBDA and shape parameter K.
     The size of X is the common size of P, LAMBDA, and K.  A scalar input
     functions as a constant matrix of the same size as the other inputs.

     Default values are LAMBDA = 1, K = 1.

     Further information about the Weibull distribution can be found at
     <https://en.wikipedia.org/wiki/Weibull_distribution>

     See also: wblcdf, wblpdf, wblrnd, wblstat, wblplot.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 63
Inverse of the Weibull cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
wblpdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 719
 -- statistics: Y = wblinv (X)
 -- statistics: Y = wblinv (X, LAMBDA)
 -- statistics: Y = wblinv (X, LAMBDA, K)

     Weibull probability density function (PDF).

     For each element of X, compute the probability density function (PDF) of
     the Weibull distribution with scale parameter LAMBDA and shpe parameter K.
     The size of Y is the common size of X, LAMBDA, and K.  A scalar input
     functions as a constant matrix of the same size as the other inputs.

     Default values are LAMBDA = 1, K = 1.

     Further information about the Weibull distribution can be found at
     <https://en.wikipedia.org/wiki/Weibull_distribution>

     See also: wblcdf, wblinv, wblrnd, wblfit, wbllike, wblstat, wblplot.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 43
Weibull probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
wblrnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1178
 -- statistics: R = wblrnd (LAMBDA, K)
 -- statistics: R = wblrnd (LAMBDA, K, ROWS)
 -- statistics: R = wblrnd (LAMBDA, K, ROWS, COLS, ...)
 -- statistics: R = wblrnd (LAMBDA, K, [SZ])

     Random arrays from the Weibull distribution.

     ‘R = wblrnd (LAMBDA, K)’ returns an array of random numbers chosen from the
     Weibull distribution with scale parameter LAMBDA and shape parameter K.
     The size of R is the common size of LAMBDA and K.  A scalar input functions
     as a constant matrix of the same size as the other inputs.  Both parameters
     must be positive reals.

     When called with a single size argument, ‘wblrnd’ returns a square matrix
     with the dimension specified.  When called with more than one scalar
     argument, the first two arguments are taken as the number of rows and
     columns and any further arguments specify additional matrix dimensions.
     The size may also be specified with a row vector of dimensions, SZ.

     Further information about the Weibull distribution can be found at
     <https://en.wikipedia.org/wiki/Weibull_distribution>

     See also: wblcdf, wblinv, wblpdf, wblfit, wbllike, wblstat, wblplot.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 44
Random arrays from the Weibull distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
wienrnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 456
 -- statistics: R = wienrnd (T, D, N)

     Return a simulated realization of the D-dimensional Wiener Process on the
     interval [0, T].

     If D is omitted, D = 1 is used.  The first column of the return matrix
     contains time, the remaining columns contain the Wiener process.

     The optional parameter N defines the number of summands used for simulating
     the process over an interval of length 1.  If N is omitted, N = 1000 is
     used.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Return a simulated realization of the D-dimensional Wiener Process on the
int...



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
wishpdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 642
 -- statistics: Y = wishpdf (W, SIGMA, DF, LOG_Y=false)

     Compute the probability density function of the Wishart distribution

     Inputs: A P x P matrix W where to find the PDF. The P x P positive definite
     matrix SIGMA and scalar degrees of freedom parameter DF characterizing the
     Wishart distribution.  (For the density to be finite, need DF > (P - 1).)

     If the flag LOG_Y is set, return the log probability density - this helps
     avoid underflow when the numerical value of the density is very small

     Output: Y is the probability density of Wishart(SIGMA, DF) at W.

     See also: wishrnd, iwishpdf, iwishrnd.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 68
Compute the probability density function of the Wishart distribution



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
wishrnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1067
 -- statistics: [W, D] = wishrnd (SIGMA, DF, D, N=1)

     Return a random matrix sampled from the Wishart distribution with given
     parameters

     Inputs: the p x p positive definite matrix SIGMA (or the lower-triangular
     Cholesky factor D of SIGMA) and scalar degrees of freedom parameter DF.

     DF can be non-integer as long as DF > p - 1

     Output: a random p x p matrix W from the Wishart(SIGMA, DF) distribution.
     If N > 1, then W is P x P x N and holds N such random matrices.
     (Optionally, the lower-triangular Cholesky factor D of SIGMA is also
     returned.)

     Averaged across many samples, the mean of W should approach DF*SIGMA, and
     the variance of each element W_ij should approach DF*(SIGMA_ij^2 +
     SIGMA_ii*SIGMA_jj)

     References
     ----------

       1. Yu-Cheng Ku and Peter Bloomfield (2010), Generating Random Wishart
          Matrices with Fractional Degrees of Freedom in OX,
          http://www.gwu.edu/~forcpgm/YuChengKu-030510final-WishartYu-ChengKu.pdf

     See also: wishpdf, iwishpdf, iwishrnd.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Return a random matrix sampled from the Wishart distribution with given
param...





