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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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