info octave

25.6 Distributions

Octave has functions for computing the Probability Density Function (PDF), the Cumulative Distribution function (CDF), and the quantile (the inverse of the CDF) of a large number of distributions.

The following table summarizes the supported distributions (in alphabetical order).

Distribution	PDF	CDF	Quantile
Beta Distribution	`betapdf`	`betacdf`	`betainv`
Binomial Distribution	`binopdf`	`binocdf`	`binoinv`
Cauchy Distribution	`cauchy_pdf`	`cauchy_cdf`	`cauchy_inv`
Chi-Square Distribution	`chi2pdf`	`chi2cdf`	`chi2inv`
Univariate Discrete Distribution	`discrete_pdf`	`discrete_cdf`	`discrete_inv`
Empirical Distribution	`empirical_pdf`	`empirical_cdf`	`empirical_inv`
Exponential Distribution	`exppdf`	`expcdf`	`expinv`
F Distribution	`fpdf`	`fcdf`	`finv`
Gamma Distribution	`gampdf`	`gamcdf`	`gaminv`
Geometric Distribution	`geopdf`	`geocdf`	`geoinv`
Hypergeometric Distribution	`hygepdf`	`hygecdf`	`hygeinv`
Kolmogorov Smirnov Distribution	Not Available	`kolmogorov_smirnov_cdf`	Not Available
Laplace Distribution	`laplace_pdf`	`laplace_cdf`	`laplace_inv`
Logistic Distribution	`logistic_pdf`	`logistic_cdf`	`logistic_inv`
Log-Normal Distribution	`lognpdf`	`logncdf`	`logninv`
Pascal Distribution	`nbinpdf`	`nbincdf`	`nbininv`
Univariate Normal Distribution	`normpdf`	`normcdf`	`norminv`
Poisson Distribution	`poisspdf`	`poisscdf`	`poissinv`
t (Student) Distribution	`tpdf`	`tcdf`	`tinv`
Univariate Discrete Distribution	`unidpdf`	`unidcdf`	`unidinv`
Uniform Distribution	`unifpdf`	`unifcdf`	`unifinv`
Weibull Distribution	`wblpdf`	`wblcdf`	`wblinv`

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: 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: 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: 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: 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: 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: expcdf (x, lambda)

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

Function File: exppdf (x, lambda): For each element of x, compute the probability density function (PDF) of the exponential distribution with mean 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: 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: 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: hygecdf (x, t, m, n)

Compute the cumulative distribution function (CDF) at x of the hypergeometric distribution with parameters t, m, 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 t, m, and n must positive integers with m and n not greater than t.

Function File: hygeinv (x, t, m, n)

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

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

Function File: hygepdf (x, t, m, n)

Compute the probability density function (PDF) at x of the hypergeometric distribution with parameters t, m, 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: 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: 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: 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: 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: 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: 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: tcdf (x, n): For each element of x, compute the cumulative distribution function (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 probability value x, compute the inverse of the cumulative distribution function (CDF) of the t (Student) distribution with degrees of freedom n. This function is analogous to looking in a table for the t-value of a single-tailed distribution.

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: unidcdf (x, v): 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 equal probability.

Function File: unidinv (x, v): For each component of x, compute the quantile (the inverse of the CDF) at x of the univariate discrete distribution which assumes the values in v with equal probability

Function File: unidpdf (x, v): 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 equal probability.

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: 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.

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