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26.4 Distributions

— Function File: betacdf (x, a, b)

For each element of x, returns the CDF at x of the beta distribution with parameters a and b, i.e., PROB (beta (a, b) <= x).

— Function File: betainv (x, a, b)

For each component of x, compute the quantile (the inverse of the CDF) at x of the Beta distribution with parameters a and b.

— Function File: betapdf (x, a, b)

For each element of x, returns the PDF at x of the beta distribution with parameters a and b.

— Function File: betarnd (a, b, r, c)
— Function File: betarnd (a, b, sz)

Return an r by c or size (sz) matrix of random samples from the Beta distribution with parameters a and b. Both a and b must be scalar or of size r by c.

If r and c are omitted, the size of the result matrix is the common size of a and b.

— Function File: binocdf (x, n, p)

For each element of x, compute the CDF at x of the binomial distribution with parameters n and p.

— Function File: binoinv (x, n, p)

For each element of x, compute the quantile at x of the binomial distribution with parameters n and p.

— Function File: binopdf (x, n, p)

For each element of x, compute the probability density function (PDF) at x of the binomial distribution with parameters n and p.

— Function File: binornd (n, p, r, c)
— Function File: binornd (n, p, sz)

Return an r by c or a size (sz) matrix of random samples from the binomial distribution with parameters n and p. Both n and p must be scalar or of size r by c.

If r and c are omitted, the size of the result matrix is the common size of n and p.

— Function File: cauchy_cdf (x, lambda, sigma)

For each element of x, compute the cumulative distribution function (CDF) at x of the Cauchy distribution with location parameter lambda and scale parameter sigma. Default values are lambda = 0, sigma = 1.

— Function File: cauchy_inv (x, lambda, sigma)

For each element of x, compute the quantile (the inverse of the CDF) at x of the Cauchy distribution with location parameter lambda and scale parameter sigma. Default values are lambda = 0, sigma = 1.

— Function File: cauchy_pdf (x, lambda, sigma)

For each element of x, compute the probability density function (PDF) at x of the Cauchy distribution with location parameter lambda and scale parameter sigma > 0. Default values are lambda = 0, sigma = 1.

— Function File: cauchy_rnd (lambda, sigma, r, c)
— Function File: cauchy_rnd (lambda, sigma, sz)

Return an r by c or a size (sz) matrix of random samples from the Cauchy distribution with parameters lambda and sigma which must both be scalar or of size r by c.

If r and c are omitted, the size of the result matrix is the common size of lambda and sigma.

— Function File: chi2cdf (x, n)

For each element of x, compute the cumulative distribution function (CDF) at x of the chisquare distribution with n degrees of freedom.

— Function File: chi2inv (x, n)

For each element of x, compute the quantile (the inverse of the CDF) at x of the chisquare distribution with n degrees of freedom.

— Function File: chisquare_pdf (x, n)

For each element of x, compute the probability density function (PDF) at x of the chisquare distribution with n degrees of freedom.

— Function File: chi2rnd (n, r, c)
— Function File: chi2rnd (n, sz)

Return an r by c or a size (sz) matrix of random samples from the chisquare distribution with n degrees of freedom. n must be a scalar or of size r by c.

If r and c are omitted, the size of the result matrix is the size of n.

— Function File: discrete_cdf (x, v, p)

For each element of x, compute the cumulative distribution function (CDF) at x of a univariate discrete distribution which assumes the values in v with probabilities p.

— Function File: discrete_inv (x, v, p)

For each component of x, compute the quantile (the inverse of the CDF) at x of the univariate distribution which assumes the values in v with probabilities p.

— Function File: discrete_pdf (x, v, p)

For each element of x, compute the probability density function (pDF) at x of a univariate discrete distribution which assumes the values in v with probabilities p.

— Function File: discrete_rnd (n, v, p)
— Function File: discrete_rnd (v, p, r, c)
— Function File: discrete_rnd (v, p, sz)

Generate a row vector containing a random sample of size n from the univariate distribution which assumes the values in v with probabilities p. n must be a scalar.

If r and c are given create a matrix with r rows and c columns. Or if sz is a vector, create a matrix of size sz.

— Function File: empirical_cdf (x, data)

For each element of x, compute the cumulative distribution function (CDF) at x of the empirical distribution obtained from the univariate sample data.

— Function File: empirical_inv (x, data)

For each element of x, compute the quantile (the inverse of the CDF) at x of the empirical distribution obtained from the univariate sample data.

— Function File: empirical_pdf (x, data)

For each element of x, compute the probability density function (PDF) at x of the empirical distribution obtained from the univariate sample data.

— Function File: empirical_rnd (n, data)
— Function File: empirical_rnd (data, r, c)
— Function File: empirical_rnd (data, sz)

Generate a bootstrap sample of size n from the empirical distribution obtained from the univariate sample data.

If r and c are given create a matrix with r rows and c columns. Or if sz is a vector, create a matrix of size sz.

— Function File: expcdf (x, lambda)

For each element of x, compute the cumulative distribution function (CDF) at x of the exponential distribution with parameter lambda.

The arguments can be of common size or scalar.

— Function File: expinv (x, lambda)

For each element of x, compute the quantile (the inverse of the CDF) at x of the exponential distribution with parameter lambda.

— Function File: exppdf (x, lambda)

For each element of x, compute the probability density function (PDF) of the exponential distribution with parameter lambda.

— Function File: exprnd (lambda, r, c)
— Function File: exprnd (lambda, sz)

Return an r by c matrix of random samples from the exponential distribution with parameter lambda, which must be a scalar or of size r by c. Or if sz is a vector, create a matrix of size sz.

If r and c are omitted, the size of the result matrix is the size of lambda.

— Function File: fcdf (x, m, n)

For each element of x, compute the CDF at x of the F distribution with m and n degrees of freedom, i.e., PROB (F (m, n) <= x).

— Function File: finv (x, m, n)

For each component of x, compute the quantile (the inverse of the CDF) at x of the F distribution with parameters m and n.

— Function File: fpdf (x, m, n)

For each element of x, compute the probability density function (PDF) at x of the F distribution with m and n degrees of freedom.

— Function File: frnd (m, n, r, c)
— Function File: frnd (m, n, sz)

Return an r by c matrix of random samples from the F distribution with m and n degrees of freedom. Both m and n must be scalar or of size r by c. If sz is a vector the random samples are in a matrix of size sz.

If r and c are omitted, the size of the result matrix is the common size of m and n.

— Function File: gamcdf (x, a, b)

For each element of x, compute the cumulative distribution function (CDF) at x of the Gamma distribution with parameters a and b.

     
     
See also: gamma, gammaln, gammainc, gampdf, gaminv, gamrnd.

— Function File: gaminv (x, a, b)

For each component of x, compute the quantile (the inverse of the CDF) at x of the Gamma distribution with parameters a and b.

     
     
See also: gamma, gammaln, gammainc, gampdf, gamcdf, gamrnd.

— Function File: gampdf (x, a, b)

For each element of x, return the probability density function (PDF) at x of the Gamma distribution with parameters a and b.

     
     
See also: gamma, gammaln, gammainc, gamcdf, gaminv, gamrnd.

— Function File: gamrnd (a, b, r, c)
— Function File: gamrnd (a, b, sz)

Return an r by c or a size (sz) matrix of random samples from the Gamma distribution with parameters a and b. Both a and b must be scalar or of size r by c.

If r and c are omitted, the size of the result matrix is the common size of a and b.

     
     
See also: gamma, gammaln, gammainc, gampdf, gamcdf, gaminv.

— Function File: geocdf (x, p)

For each element of x, compute the CDF at x of the geometric distribution with parameter p.

— Function File: geoinv (x, p)

For each element of x, compute the quantile at x of the geometric distribution with parameter p.

— Function File: geopdf (x, p)

For each element of x, compute the probability density function (PDF) at x of the geometric distribution with parameter p.

— Function File: geornd (p, r, c)
— Function File: geornd (p, sz)

Return an r by c matrix of random samples from the geometric distribution with parameter p, which must be a scalar or of size r by c.

If r and c are given create a matrix with r rows and c columns. Or if sz is a vector, create a matrix of size sz.

— Function File: hygecdf (x, m, t, n)

Compute the cumulative distribution function (CDF) at x of the hypergeometric distribution with parameters m, t, and n. This is the 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 t containing m marked items.

The parameters m, t, and n must positive integers with m and n not greater than t.

— Function File: hygeinv (x, m, t, n)

For each element of x, compute the quantile at x of the hypergeometric distribution with parameters m, t, and n.

The parameters m, t, and n must positive integers with m and n not greater than t.

— Function File: hygepdf (x, m, t, n)

Compute the probability density function (PDF) at x of the hypergeometric distribution with parameters m, t, and n. 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 t containing m marked items.

The arguments must be of common size or scalar.

— Function File: hygernd (m, t, n, r, c)
— Function File: hygernd (m, t, n, sz)

Return an r by c matrix of random samples from the hypergeometric distribution with parameters m, t, and n.

The parameters m, t, and n must positive integers with m and n not greater than t.

— Function File: kolmogorov_smirnov_cdf (x, tol)

Return the CDF at x of the Kolmogorov-Smirnov distribution,

                   Inf
          Q(x) =   SUM    (-1)^k exp(-2 k^2 x^2)
                 k = -Inf
     

for x > 0.

The optional parameter tol specifies the precision up to which the series should be evaluated; the default is tol = eps.

— Function File: laplace_cdf (x)

For each element of x, compute the cumulative distribution function (CDF) at x of the Laplace distribution.

— Function File: laplace_inv (x)

For each element of x, compute the quantile (the inverse of the CDF) at x of the Laplace distribution.

— Function File: laplace_pdf (x)

For each element of x, compute the probability density function (PDF) at x of the Laplace distribution.

— Function File: laplace_rnd (r, c)
— Function File: laplace_rnd (sz);

Return an r by c matrix of random numbers from the Laplace distribution. Or if sz is a vector, create a matrix of sz.

— Function File: logistic_cdf (x)

For each component of x, compute the CDF at x of the logistic distribution.

— Function File: logistic_inv (x)

For each component of x, compute the quantile (the inverse of the CDF) at x of the logistic distribution.

— Function File: logistic_pdf (x)

For each component of x, compute the PDF at x of the logistic distribution.

— Function File: logistic_rnd (r, c)
— Function File: logistic_rnd (sz)

Return an r by c matrix of random numbers from the logistic distribution. Or if sz is a vector, create a matrix of sz.

— Function File: logncdf (x, mu, sigma)

For each element of x, compute the cumulative distribution function (CDF) at x of the lognormal distribution with parameters mu and sigma. If a random variable follows this distribution, its logarithm is normally distributed with mean mu and standard deviation sigma.

Default values are mu = 1, sigma = 1.

— Function File: logninv (x, mu, sigma)

For each element of x, compute the quantile (the inverse of the CDF) at x of the lognormal distribution with parameters mu and sigma. If a random variable follows this distribution, its logarithm is normally distributed with mean log (mu) and variance sigma.

Default values are mu = 1, sigma = 1.

— Function File: lognpdf (x, mu, sigma)

For each element of x, compute the probability density function (PDF) at x of the lognormal distribution with parameters mu and sigma. If a random variable follows this distribution, its logarithm is normally distributed with mean mu and standard deviation sigma.

Default values are mu = 1, sigma = 1.

— Function File: lognrnd (mu, sigma, r, c)
— Function File: lognrnd (mu, sigma, sz)

Return an r by c matrix of random samples from the lognormal distribution with parameters mu and sigma. Both mu and sigma must be scalar or of size r by c. Or if sz is a vector, create a matrix of size sz.

If r and c are omitted, the size of the result matrix is the common size of mu and sigma.

— Function File: nbincdf (x, n, p)

For each element of x, compute the CDF at x of the Pascal (negative binomial) distribution with parameters n and p.

The number of failures in a Bernoulli experiment with success probability p before the n-th success follows this distribution.

— Function File: nbininv (x, n, p)

For each element of x, compute the quantile at x of the Pascal (negative binomial) distribution with parameters n and p.

The number of failures in a Bernoulli experiment with success probability p before the n-th success follows this distribution.

— Function File: nbinpdf (x, n, p)

For each element of x, compute the probability density function (PDF) at x of the Pascal (negative binomial) distribution with parameters n and p.

The number of failures in a Bernoulli experiment with success probability p before the n-th success follows this distribution.

— Function File: nbinrnd (n, p, r, c)
— Function File: nbinrnd (n, p, sz)

Return an r by c matrix of random samples from the Pascal (negative binomial) distribution with parameters n and p. Both n and p must be scalar or of size r by c.

If r and c are omitted, the size of the result matrix is the common size of n and p. Or if sz is a vector, create a matrix of size sz.

— Function File: normcdf (x, m, s)

For each element of x, compute the cumulative distribution function (CDF) at x of the normal distribution with mean m and standard deviation s.

Default values are m = 0, s = 1.

— Function File: norminv (x, m, s)

For each element of x, compute the quantile (the inverse of the CDF) at x of the normal distribution with mean m and standard deviation s.

Default values are m = 0, s = 1.

— Function File: normpdf (x, m, s)

For each element of x, compute the probability density function (PDF) at x of the normal distribution with mean m and standard deviation s.

Default values are m = 0, s = 1.

— Function File: normrnd (m, s, r, c)
— Function File: normrnd (m, s, sz)

Return an r by c or size (sz) matrix of random samples from the normal distribution with parameters mean m and standard deviation s. Both m and s must be scalar or of size r by c.

If r and c are omitted, the size of the result matrix is the common size of m and s.

— Function File: poisscdf (x, lambda)

For each element of x, compute the cumulative distribution function (CDF) at x of the Poisson distribution with parameter lambda.

— Function File: poissinv (x, lambda)

For each component of x, compute the quantile (the inverse of the CDF) at x of the Poisson distribution with parameter lambda.

— Function File: poisspdf (x, lambda)

For each element of x, compute the probability density function (PDF) at x of the poisson distribution with parameter lambda.

— Function File: poissrnd (lambda, r, c)

Return an r by c matrix of random samples from the Poisson distribution with parameter lambda, which must be a scalar or of size r by c.

If r and c are omitted, the size of the result matrix is the size of lambda.

— Function File: tcdf (x, n)

For each element of x, compute the CDF at x of the t (Student) distribution with n degrees of freedom, i.e., PROB (t(n) <= x).

— Function File: tinv (x, n)

For each component of x, compute the quantile (the inverse of the CDF) at x of the t (Student) distribution with parameter n.

— Function File: tpdf (x, n)

For each element of x, compute the probability density function (PDF) at x of the t (Student) distribution with n degrees of freedom.

— Function File: trnd (n, r, c)
— Function File: trnd (n, sz)

Return an r by c matrix of random samples from the t (Student) distribution with n degrees of freedom. n must be a scalar or of size r by c. Or if sz is a vector create a matrix of size sz.

If r and c are omitted, the size of the result matrix is the size of n.

— Function File: unidrnd (mx);
— Function File: unidrnd (mx, v);
— Function File: unidrnd (mx, m, n, ...);

Return random values from discrete uniform distribution, with maximum value(s) given by the integer mx, which may be a scalar or multidimensional array.

If mx is a scalar, the size of the result is specified by the vector v, or by the optional arguments m, n, .... Otherwise, the size of the result is the same as the size of mx.

— Function File: unifcdf (x, a, b)

Return the CDF at x of the uniform distribution on [a, b], i.e., PROB (uniform (a, b) <= x).

Default values are a = 0, b = 1.

— Function File: unifinv (x, a, b)

For each element of x, compute the quantile (the inverse of the CDF) at x of the uniform distribution on [a, b].

Default values are a = 0, b = 1.

— Function File: unifpdf (x, a, b)

For each element of x, compute the PDF at x of the uniform distribution on [a, b].

Default values are a = 0, b = 1.

— Function File: unifrnd (a, b, r, c)
— Function File: unifrnd (a, b, sz)

Return an r by c or a size (sz) matrix of random samples from the uniform distribution on [a, b]. Both a and b must be scalar or of size r by c.

If r and c are omitted, the size of the result matrix is the common size of a and b.

— Function File: wblcdf (x, scale, shape)

Compute the cumulative distribution function (CDF) at x of the Weibull distribution with shape parameter scale and scale parameter shape, which is

          1 - exp(-(x/shape)^scale)
     

for x >= 0.

— Function File: wblinv (x, scale, shape)

Compute the quantile (the inverse of the CDF) at x of the Weibull distribution with shape parameter scale and scale parameter shape.

— Function File: wblpdf (x, scale, shape)

Compute the probability density function (PDF) at x of the Weibull distribution with shape parameter scale and scale parameter shape which is given by

             scale * shape^(-scale) * x^(scale-1) * exp(-(x/shape)^scale)
     

for x > 0.

— Function File: wblrnd (scale, shape, r, c)
— Function File: wblrnd (scale, shape, sz)

Return an r by c matrix of random samples from the Weibull distribution with parameters scale and shape which must be scalar or of size r by c. Or if sz is a vector return a matrix of size sz.

If r and c are omitted, the size of the result matrix is the common size of alpha and sigma.

— Function File: 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 gives the number of summands used for simulating the process over an interval of length 1. If n is omitted, n = 1000 is used.