GSL.Random.Dist

Random number distributions. Functions for generating random variates and computing their probability distributions.

The Gaussian Distribution

General

gaussianPdf :: Double -> Double -> Double

gaussianPdf x sigma computes the probabililty density p(x) for a Gaussian distribution with mean 0 and standard deviation sigma.

gaussianP :: Double -> Double -> Double

gaussianP x sigma computes the cumulative distribution function P(x) for a Gaussian distribution with mean 0 and standard deviation sigma.

gaussianQ :: Double -> Double -> Double

gaussianQ x sigma computes the cumulative distribution function Q(x) for a Gaussian distribution with mean 0 and standard deviation sigma.

gaussianPInv :: Double -> Double -> Double

gaussianPInv p sigma computes the inverse of the cumulative distribution function of a Gaussian distribution with mean 0 and standard deviation sigma. It returns x such that P(x) = p.

gaussianQInv :: Double -> Double -> Double

gaussianPInv q sigma computes the inverse of the cumulative distribution function of a Gaussian distribution with mean 0 and standard deviation sigma. It returns x such that Q(x) = q.

getGaussian :: RNG -> Double -> IO Double

getGaussian r sigma gets a normal random variable with mean 0 and standard deviation sigma. This uses the Box-Mueller algorithm.

getGaussianZiggurat :: RNG -> Double -> IO Double

getGaussianZiggurat r sigma gets a normal random variable with mean 0 and standard deviation sigma. This uses the Marsaglia-Tsang ziggurat algorithm.

getGaussianRatioMethod :: RNG -> Double -> IO Double

getGaussianRatioMethod r sigma gets a normal random variable with mean 0 and standard deviation sigma. This uses the Kinderman-Monahan-Leva ratio method.

Unit Variance

ugaussianPdf :: Double -> Double

ugaussianPdf x computes the probabililty density p(x) for a Gaussian distribution with mean 0 and standard deviation 1.

ugaussianP :: Double -> Double

ugaussianP x computes the cumulative distribution function P(x) for a Gaussian distribution with mean 0 and standard deviation 1.

ugaussianQ :: Double -> Double

ugaussianQ x computes the cumulative distribution function Q(x) for a Gaussian distribution with mean 0 and standard deviation 1.

ugaussianPInv :: Double -> Double

ugaussianPInv p computes the inverse of the cumulative distribution function of a Gaussian distribution with mean 0 and standard deviation 1. It returns x such that P(x) = p.

ugaussianQInv :: Double -> Double

ugaussianPInv q computes the inverse of the cumulative distribution function of a Gaussian distribution with mean 0 and standard deviation 1. It returns x such that Q(x) = q.

getUGaussian :: RNG -> IO Double

getUGaussian r gets a normal random variable with mean 0 and standard deviation 1. This uses the Box-Mueller algorithm.

getUGaussianRatioMethod :: RNG -> IO Double

getUGaussianRatioMethod r gets a normal random variable with mean 0 and standard deviation 1. This uses the Kinderman-Monahan-Leva ratio method.

The Flat (Uniform) Distribution

flatPdf :: Double -> Double -> Double -> Double

flatPdf x a b computes the probability density p(x) at x for a uniform distribution from a to b.

flatP :: Double -> Double -> Double -> Double

flatP x a b computes the cumulative distribution function P(x).

flatQ :: Double -> Double -> Double -> Double

flatQ x a b computes the cumulative distribution function Q(x).

flatPInv :: Double -> Double -> Double -> Double

flatPInv p a b computes the inverse of the cumulative distribution and returns x so that function P(x) = p.

flatQInv :: Double -> Double -> Double -> Double

flatQInv q a b computes the inverse of the cumulative distribution and returns x so that function Q(x) = q.

getFlat :: RNG -> Double -> Double -> IO Double

getFlat r a b gets a value uniformly chosen in [a,b).

The Exponential Distribution

exponentialPdf :: Double -> Double -> Double

exponentialPdf x mu computes the density at x of an exponential with mean mu.

exponentialP :: Double -> Double -> Double

exponentialQ :: Double -> Double -> Double

exponentialPInv :: Double -> Double -> Double

exponentialQInv :: Double -> Double -> Double

getExponential :: RNG -> Double -> IO Double

getExponential r mu gets a random exponential with mean mu.

The Levy alpha-Stable Distributions

getLevy :: RNG -> Double -> Double -> IO Double

getLevy r c alpha gets a variate from the Levy symmetric stable distribution with scale c and exponent alpha. The algorithm only works for 0 <= alpha <= 2.

getLevySkew :: RNG -> Double -> Double -> Double -> IO Double

getLevySkew r c alpha beta gets a variate from the Levy skew stable distribution with scale c, exponent alpha, and skewness parameter beta. The skewness parameter must lie in the range [-1,1]. The algorithm only works for 0 <= alpha <= 2.

The Poisson Distribution

poissonPdf :: Int -> Double -> Double

poissonPdf k mu evaluates the probability density p(k) at k for a Poisson distribution with mean mu.

poissonP :: Int -> Double -> Double

poissonP k mu evaluates the cumulative distribution function P(k) at k for a Poisson distribution with mean mu.

poissonQ :: Int -> Double -> Double

poissonQ k mu evaluates the cumulative distribution function Q(k) at k for a Poisson distribution with mean mu.

getPoisson :: RNG -> Double -> IO Int

getPoisson r mu gets a poisson random variable with mean mu.

The Cauchy Distribution

getCauchy :: RNG -> Double -> IO Double

getCauchy r a gets a random cauchy with scale a.

cauchyPdf :: Double -> Double -> Double

cauchyPdf x a evaluates the probability density p(x) at x for a Cauchy distribution with scale parameter a. The density is given by p(x) dx = { 1 over api (1 + (x/a^2)) } dx.

cauchyP :: Double -> Double -> Double

cauchyQ :: Double -> Double -> Double

cauchyPInv :: Double -> Double -> Double

cauchyQInv :: Double -> Double -> Double

The Beta Distribution

getBeta :: RNG -> Double -> Double -> IO Double

getBeta r a b gets a random beta with parameters a and b.

betaPdf :: Double -> Double -> Double -> Double

betaPdf x a b evaluates the probability density p(x) at x for a Beta distribution with parameters a and b. The density is given by p(x) dx = {Gamma(a+b) over Gamma(a) Gamma(b)} x^{a-1} (1-x)^{b-1} dx for 0 <= x <= 1.

betaP :: Double -> Double -> Double -> Double

betaQ :: Double -> Double -> Double -> Double

betaPInv :: Double -> Double -> Double -> Double

betaQInv :: Double -> Double -> Double -> Double

The Logistic Distribution

getLogistic :: RNG -> Double -> IO Double

getLogistic r a gets a random logistic with scale a.

logisticPdf :: Double -> Double -> Double

logisticPdf x a evaluates the probability density p(x) at x for a logistic distribution with scale parameter a. The density is given by p(x) dx = { exp(-xa) \over a (1 + \exp(-xa))^2 } dx.

logisticP :: Double -> Double -> Double

logisticQ :: Double -> Double -> Double

logisticPInv :: Double -> Double -> Double

logisticQInv :: Double -> Double -> Double

The Log-Normal Distribution

getLognormal :: RNG -> Double -> Double -> IO Double

getLognormal zeta sigma gets a random lognormal with parameters zeta and sigma.

lognormalPdf :: Double -> Double -> Double -> Double

lognormalPdf x zeta sigma evaluates the probability density p(x) at x for a log-normal distribution with parameters zeta and sigma, given. The density is given by p(x) dx = p(x) {1 over x sqrt{2 pi sigma^2} } exp(-(ln(x) - zeta)^2/2 sigma^2) dx

lognormalP :: Double -> Double -> Double -> Double

lognormalQ :: Double -> Double -> Double -> Double

lognormalPInv :: Double -> Double -> Double -> Double

lognormalQInv :: Double -> Double -> Double -> Double

The Pareto Distribution

getPareto :: RNG -> Double -> Double -> IO Double

getPareto r a b gets a random Pareto with exponent a and scale b.

paretoPdf :: Double -> Double -> Double -> Double

paretoPdf x a b evaluates the probability density p(x) at x for a Pareto distribution with exponent a and scale b. The density is given by p(x) dx = (ab) (x/b)^{a+1} dx for x >= b.

paretoP :: Double -> Double -> Double -> Double

paretoQ :: Double -> Double -> Double -> Double

paretoPInv :: Double -> Double -> Double -> Double

paretoQInv :: Double -> Double -> Double -> Double

The Weibull Distribution

getWeibull :: RNG -> Double -> Double -> IO Double

getWeibull r a b gets a random Weibull with scale a and exponent b.

weibullPdf :: Double -> Double -> Double -> Double

weibullPdf x a b evaluates the probability density p(x) at x for a Weibull distribution with scale a and exponent b. The density is given by p(x) dx = {b over a^b} x^{b-1} exp(-(x/a)^b) dx for x >= 0.

weibullP :: Double -> Double -> Double -> Double

weibullQ :: Double -> Double -> Double -> Double

weibullPInv :: Double -> Double -> Double -> Double

weibullQInv :: Double -> Double -> Double -> Double

The Gamma Distribution

getGamma :: RNG -> Double -> Double -> IO Double

getGamma r a b gets a random gamma with parameters a and b. Uses the Marsagli-Tsang fast gamma method.

getGammaKnuth :: RNG -> Double -> Double -> IO Double

getGammaKnuth r a b gets a random gamma with parameters a and b. Uses the algorithms from Knuth (vol 2).

gammaPdf :: Double -> Double -> Double -> Double

gammaPdf x a b evaluates the probability density p(x) at x for a gamma distribution with parameters a and b. The density is given by p(x) dx = p(x) dx = {1 over Gamma(a) b^a} x^{a-1} e^{-x/b} dx for x > 0.

gammaP :: Double -> Double -> Double -> Double

gammaQ :: Double -> Double -> Double -> Double

gammaPInv :: Double -> Double -> Double -> Double

gammaQInv :: Double -> Double -> Double -> Double

The Multinomial Distribution

getMultinomial :: RNG -> Int -> Vector Double -> IO (Vector Int)

getMultinomial r n ps gets a random sample from a multinomial distribution with parameters ps formed by n trials.

multinomialPdf

Arguments

:: Vector Int

ns

-> Vector Double

ps

-> Double

multinomialPdf ns ps evaluates the probability density p(ns) at ns for a multinomial distribution with parameters ps, where all ps are non-negative and sum to 1. Note that xs and alphas should have the same length.

multinomialLnPdf

Arguments

:: Vector Int

ns

-> Vector Double

ps

-> Double

multinomialLnPdf xs alphas == log (multinomialPdf xs alphas), but more efficient.

The Dirichlet Distribution

getDirichlet :: RNG -> Vector Double -> IO (Vector Double)

getDirichlet r alphas gets a random sample from a Dirichlet distribution with parameters alphas, where all alphas are positive.

dirichletPdf

Arguments

:: Vector Double

xs

-> Vector Double

alphas

-> Double

dirichletPdf xs alphas evaluates the probability density p(xs) at xs for a Dirichlet distribution with parameters alphas, where all alphas are positive (strictly greater than zero). Note that xs and alphas should have the same length.

dirichletLnPdf

Arguments

:: Vector Double

xs

-> Vector Double

alphas

-> Double

dirichletLnPdf xs alphas == log (dirichletPdf xs alphas), but more efficient.