Univariate Distributions

# Univariate Distributions

Univariate distributions are the distributions whose variate forms are Univariate (i.e each sample is a scalar). Abstract types for univariate distributions:

const UnivariateDistribution{S<:ValueSupport} = Distribution{Univariate,S}

const DiscreteUnivariateDistribution   = Distribution{Univariate, Discrete}
const ContinuousUnivariateDistribution = Distribution{Univariate, Continuous}

## Common Interface

A series of methods are implemented for each univariate distribution, which provide useful functionalities such as moment computation, pdf evaluation, and sampling (i.e. random number generation).

### Parameter Retrieval

Note: params are defined for all univariate distributions, while other parameter retrieval methods are only defined for those distributions for which these parameters make sense. See below for details.

params(d::UnivariateDistribution)

Return a tuple of parameters. Let d be a distribution of type D, then D(params(d)...) will construct exactly the same distribution as $d$.

scale(d::UnivariateDistribution)

Get the scale parameter.

location(d::UnivariateDistribution)

Get the location parameter.

shape(d::UnivariateDistribution)

Get the shape parameter.

rate(d::UnivariateDistribution)

Get the rate parameter.

ncategories(d::UnivariateDistribution)

Get the number of categories.

ntrials(d::UnivariateDistribution)

Get the number of trials.

dof(d::UnivariateDistribution)

Get the degrees of freedom.

For distributions for which success and failure have a meaning, the following methods are defined:

succprob(d::DiscreteUnivariateDistribution)

Get the probability of success.

failprob(d::DiscreteUnivariateDistribution)

Get the probability of failure.

### Computation of statistics

maximum(d::UnivariateDistribution)

Return the maximum of the support of d.

minimum(d::UnivariateDistribution)

Return the minimum of the support of d.

extrema(d::UnivariateDistribution)

Return the minimum and maximum of the support of d as a 2-tuple.

mean(d::UnivariateDistribution)

Compute the expectation.

var(d::UnivariateDistribution)

Compute the variance. (A generic std is provided as std(d) = sqrt(var(d)))

std(d::UnivariateDistribution)

Return the standard deviation of distribution d, i.e. sqrt(var(d)).

median(d::UnivariateDistribution)

Return the median value of distribution d.

modes(d::UnivariateDistribution)

Get all modes (if this makes sense).

mode(d::UnivariateDistribution)

Returns the first mode.

skewness(d::UnivariateDistribution)

Compute the skewness.

kurtosis(d::UnivariateDistribution)

Compute the excessive kurtosis.

kurtosis(d::Distribution, correction::Bool)

Computes excess kurtosis by default. Proper kurtosis can be returned with correction=false

isplatykurtic(d)

Return whether d is platykurtic (i.e kurtosis(d) < 0).

isleptokurtic(d)

Return whether d is leptokurtic (i.e kurtosis(d) > 0).

ismesokurtic(d)

Return whether d is mesokurtic (i.e kurtosis(d) == 0).

entropy(d::UnivariateDistribution)

Compute the entropy value of distribution d.

entropy(d::UnivariateDistribution, b::Real)

Compute the entropy value of distribution d, w.r.t. a given base.

entropy(d::UnivariateDistribution, b::Real)

Compute the entropy value of distribution d, w.r.t. a given base.

mgf(d::UnivariateDistribution, t)

Evaluate the moment generating function of distribution d.

cf(d::UnivariateDistribution, t)

Evaluate the characteristic function of distribution d.

### Probability Evaluation

insupport(d::UnivariateDistribution, x::Any)

When x is a scalar, it returns whether x is within the support of d (e.g., insupport(d, x) = minimum(d) <= x <= maximum(d)). When x is an array, it returns whether every element in x is within the support of d.

Generic fallback methods are provided, but it is often the case that insupport can be done more efficiently, and a specialized insupport is thus desirable. You should also override this function if the support is composed of multiple disjoint intervals.

pdf(d::UnivariateDistribution, x::Real)

Evaluate the probability density (mass) at x.

See also: logpdf.

logpdf(d::UnivariateDistribution, x::Real)

Evaluate the logarithm of probability density (mass) at x. Whereas there is a fallback implemented logpdf(d, x) = log(pdf(d, x)). Relying on this fallback is not recommended in general, as it is prone to overflow or underflow.

loglikelihood(d::UnivariateDistribution, X::AbstractArray)

The log-likelihood of distribution d w.r.t. all samples contained in array x.

cdf(d::UnivariateDistribution, x::Real)

Evaluate the cumulative probability at x.

See also ccdf, logcdf, and logccdf.

logcdf(d::UnivariateDistribution, x::Real)

The logarithm of the cumulative function value(s) evaluated at x, i.e. log(cdf(x)).

logdiffcdf(d::UnivariateDistribution, x::T, y::T) where {T <: Real}

The natural logarithm of the difference between the cumulative density function at x and y, i.e. log(cdf(x) - cdf(y)).

ccdf(d::UnivariateDistribution, x::Real)

The complementary cumulative function evaluated at x, i.e. 1 - cdf(d, x).

logccdf(d::UnivariateDistribution, x::Real)

The logarithm of the complementary cumulative function values evaluated at x, i.e. log(ccdf(x)).

quantile(d::UnivariateDistribution, q::Real)

Evaluate the inverse cumulative distribution function at q.

See also: cquantile, invlogcdf, and invlogccdf.

cquantile(d::UnivariateDistribution, q::Real)

The complementary quantile value, i.e. quantile(d, 1-q).

invlogcdf(d::UnivariateDistribution, lp::Real)

The inverse function of logcdf.

invlogccdf(d::UnivariateDistribution, lp::Real)

The inverse function of logccdf.

### Sampling (Random number generation)

rand(rng::AbstractRNG, d::UnivariateDistribution)

Generate a scalar sample from d. The general fallback is quantile(d, rand()).

rand!(rng::AbstractRNG, ::UnivariateDistribution, ::AbstractArray)

Sample a univariate distribution and store the results in the provided array.

## Continuous Distributions

Arcsine(a,b)

The Arcsine distribution has probability density function

$f(x) = \frac{1}{\pi \sqrt{(x - a) (b - x)}}, \quad x \in [a, b]$
Arcsine()        # Arcsine distribution with support [0, 1]
Arcsine(b)       # Arcsine distribution with support [0, b]
Arcsine(a, b)    # Arcsine distribution with support [a, b]

params(d)        # Get the parameters, i.e. (a, b)
minimum(d)       # Get the lower bound, i.e. a
maximum(d)       # Get the upper bound, i.e. b
location(d)      # Get the left bound, i.e. a
scale(d)         # Get the span of the support, i.e. b - a

Use Arcsine(a, b, check_args=false) to bypass argument checks.

Beta(α,β)

The Beta distribution has probability density function

$f(x; \alpha, \beta) = \frac{1}{B(\alpha, \beta)} x^{\alpha - 1} (1 - x)^{\beta - 1}, \quad x \in [0, 1]$

The Beta distribution is related to the Gamma distribution via the property that if $X \sim \operatorname{Gamma}(\alpha)$ and $Y \sim \operatorname{Gamma}(\beta)$ independently, then $X / (X + Y) \sim Beta(\alpha, \beta)$.

Beta()        # equivalent to Beta(1, 1)
Beta(a)       # equivalent to Beta(a, a)
Beta(a, b)    # Beta distribution with shape parameters a and b

params(d)     # Get the parameters, i.e. (a, b)

BetaPrime(α,β)

The Beta prime distribution has probability density function

$f(x; \alpha, \beta) = \frac{1}{B(\alpha, \beta)} x^{\alpha - 1} (1 + x)^{- (\alpha + \beta)}, \quad x > 0$

The Beta prime distribution is related to the Beta distribution via the relation ship that if $X \sim \operatorname{Beta}(\alpha, \beta)$ then $\frac{X}{1 - X} \sim \operatorname{BetaPrime}(\alpha, \beta)$

BetaPrime()        # equivalent to BetaPrime(1, 1)
BetaPrime(a)       # equivalent to BetaPrime(a, a)
BetaPrime(a, b)    # Beta prime distribution with shape parameters a and b

params(d)          # Get the parameters, i.e. (a, b)

Biweight(μ, σ)
Cauchy(μ, σ)

The Cauchy distribution with location μ and scale σ has probability density function

$f(x; \mu, \sigma) = \frac{1}{\pi \sigma \left(1 + \left(\frac{x - \mu}{\sigma} \right)^2 \right)}$
Cauchy()         # Standard Cauchy distribution, i.e. Cauchy(0, 1)
Cauchy(u)        # Cauchy distribution with location u and unit scale, i.e. Cauchy(u, 1)
Cauchy(u, b)     # Cauchy distribution with location u and scale b

params(d)        # Get the parameters, i.e. (u, b)
location(d)      # Get the location parameter, i.e. u
scale(d)         # Get the scale parameter, i.e. b

Chernoff()

The Chernoff distribution is the distribution of the random variable

$\underset{t \in (-\infty,\infty)}{\arg\max} ( G(t) - t^2 ),$

where $G$ is standard two–sided Brownian motion.

The distribution arises as the limit distribution of various cube-root-n consistent estimators, including the isotonic regression estimator of Brunk, the isotonic density estimator of Grenander, the least median of squares estimator of Rousseeuw, and the maximum score estimator of Manski.

For theoretical results, see e.g. Kim and Pollard, Annals of Statistics, 1990. The code for the computation of pdf and cdf is based on the algorithm described in Groeneboom and Wellner, Journal of Computational and Graphical Statistics, 2001.

Chernoff()
pdf(Chernoff(),x::Real)
cdf(Chernoff(),x::Real)
logpdf(Chernoff(),x::Real)
survivor(Chernoff(),x::Real)
mean(Chernoff())
var(Chernoff())
skewness(Chernoff())
kurtosis(Chernoff())
kurtosis(Chernoff(), excess::Bool)
mode(Chernoff())
entropy(Chernoff())
rand(Chernoff())
rand(rng, Chernoff()
cdf(Chernoff(),-x)              #For tail probabilities, use this instead of 1-cdf(Chernoff(),x)
Chi(ν)

The Chi distribution ν degrees of freedom has probability density function

$f(x; k) = \frac{1}{\Gamma(k/2)} 2^{1 - k/2} x^{k-1} e^{-x^2/2}, \quad x > 0$

It is the distribution of the square-root of a Chisq variate.

Chi(k)       # Chi distribution with k degrees of freedom

params(d)    # Get the parameters, i.e. (k,)
dof(d)       # Get the degrees of freedom, i.e. k

Chisq(ν)

The Chi squared distribution (typically written χ²) with ν degrees of freedom has the probability density function

$f(x; k) = \frac{x^{k/2 - 1} e^{-x/2}}{2^{k/2} \Gamma(k/2)}, \quad x > 0.$

If ν is an integer, then it is the distribution of the sum of squares of ν independent standard Normal variates.

Chisq(k)     # Chi-squared distribution with k degrees of freedom

params(d)    # Get the parameters, i.e. (k,)
dof(d)       # Get the degrees of freedom, i.e. k

Cosine(μ, σ)

A raised Cosine distribution.

Epanechnikov(μ, σ)
Erlang(α,θ)

The Erlang distribution is a special case of a Gamma distribution with integer shape parameter.

Erlang()       # Erlang distribution with unit shape and unit scale, i.e. Erlang(1, 1)
Erlang(a)      # Erlang distribution with shape parameter a and unit scale, i.e. Erlang(a, 1)
Erlang(a, s)   # Erlang distribution with shape parameter a and scale s

Exponential(θ)

The Exponential distribution with scale parameter θ has probability density function

$f(x; \theta) = \frac{1}{\theta} e^{-\frac{x}{\theta}}, \quad x > 0$
Exponential()      # Exponential distribution with unit scale, i.e. Exponential(1)
Exponential(b)     # Exponential distribution with scale b

params(d)          # Get the parameters, i.e. (b,)
scale(d)           # Get the scale parameter, i.e. b
rate(d)            # Get the rate parameter, i.e. 1 / b

FDist(ν1, ν2)

The F distribution has probability density function

$f(x; \nu_1, \nu_2) = \frac{1}{x B(\nu_1/2, \nu_2/2)} \sqrt{\frac{(\nu_1 x)^{\nu_1} \cdot \nu_2^{\nu_2}}{(\nu_1 x + \nu_2)^{\nu_1 + \nu_2}}}, \quad x>0$

It is related to the Chisq distribution via the property that if $X_1 \sim \operatorname{Chisq}(\nu_1)$ and $X_2 \sim \operatorname{Chisq}(\nu_2)$, then $(X_1/\nu_1) / (X_2 / \nu_2) \sim \operatorname{FDist}(\nu_1, \nu_2)$.

FDist(ν1, ν2)     # F-Distribution with parameters ν1 and ν2

params(d)         # Get the parameters, i.e. (ν1, ν2)

Frechet(α,θ)

The Fréchet distribution with shape α and scale θ has probability density function

$f(x; \alpha, \theta) = \frac{\alpha}{\theta} \left( \frac{x}{\theta} \right)^{-\alpha-1} e^{-(x/\theta)^{-\alpha}}, \quad x > 0$
Frechet()        # Fréchet distribution with unit shape and unit scale, i.e. Frechet(1, 1)
Frechet(a)       # Fréchet distribution with shape a and unit scale, i.e. Frechet(a, 1)
Frechet(a, b)    # Fréchet distribution with shape a and scale b

params(d)        # Get the parameters, i.e. (a, b)
shape(d)         # Get the shape parameter, i.e. a
scale(d)         # Get the scale parameter, i.e. b

Gamma(α,θ)

The Gamma distribution with shape parameter α and scale θ has probability density function

$f(x; \alpha, \theta) = \frac{x^{\alpha-1} e^{-x/\theta}}{\Gamma(\alpha) \theta^\alpha}, \quad x > 0$
Gamma()          # Gamma distribution with unit shape and unit scale, i.e. Gamma(1, 1)
Gamma(α)         # Gamma distribution with shape α and unit scale, i.e. Gamma(α, 1)
Gamma(α, θ)      # Gamma distribution with shape α and scale θ

params(d)        # Get the parameters, i.e. (α, θ)
shape(d)         # Get the shape parameter, i.e. α
scale(d)         # Get the scale parameter, i.e. θ

GeneralizedExtremeValue(μ, σ, ξ)

The Generalized extreme value distribution with shape parameter ξ, scale σ and location μ has probability density function

$f(x; \xi, \sigma, \mu) = \begin{cases} \frac{1}{\sigma} \left[ 1+\left(\frac{x-\mu}{\sigma}\right)\xi\right]^{-1/\xi-1} \exp\left\{-\left[ 1+ \left(\frac{x-\mu}{\sigma}\right)\xi\right]^{-1/\xi} \right\} & \text{for } \xi \neq 0 \\\ \frac{1}{\sigma} \exp\left\{-\frac{x-\mu}{\sigma}\right\} \exp\left\{-\exp\left[-\frac{x-\mu}{\sigma}\right]\right\} & \text{for } \xi = 0 \\ \end{cases}$

for

$x \in \begin{cases} \left[ \mu - \frac{\sigma}{\xi}, + \infty \right) & \text{for } \xi > 0 \\ \left( - \infty, + \infty \right) & \text{for } \xi = 0 \\ \left( - \infty, \mu - \frac{\sigma}{\xi} \right] & \text{for } \xi < 0 \end{cases}$
GeneralizedExtremeValue(m, s, k)      # Generalized Pareto distribution with shape k, scale s and location m.

params(d)       # Get the parameters, i.e. (m, s, k)
location(d)     # Get the location parameter, i.e. m
scale(d)        # Get the scale parameter, i.e. s
shape(d)        # Get the shape parameter, i.e. k (sometimes called c)

GeneralizedPareto(μ, σ, ξ)

The Generalized Pareto distribution with shape parameter ξ, scale σ and location μ has probability density function

$f(x; \mu, \sigma, \xi) = \begin{cases} \frac{1}{\sigma}(1 + \xi \frac{x - \mu}{\sigma} )^{-\frac{1}{\xi} - 1} & \text{for } \xi \neq 0 \\ \frac{1}{\sigma} e^{-\frac{\left( x - \mu \right) }{\sigma}} & \text{for } \xi = 0 \end{cases}~, \quad x \in \begin{cases} \left[ \mu, \infty \right] & \text{for } \xi \geq 0 \\ \left[ \mu, \mu - \sigma / \xi \right] & \text{for } \xi < 0 \end{cases}$
GeneralizedPareto()             # Generalized Pareto distribution with unit shape and unit scale, i.e. GeneralizedPareto(0, 1, 1)
GeneralizedPareto(k, s)         # Generalized Pareto distribution with shape k and scale s, i.e. GeneralizedPareto(0, k, s)
GeneralizedPareto(m, k, s)      # Generalized Pareto distribution with shape k, scale s and location m.

params(d)       # Get the parameters, i.e. (m, s, k)
location(d)     # Get the location parameter, i.e. m
scale(d)        # Get the scale parameter, i.e. s
shape(d)        # Get the shape parameter, i.e. k

Gumbel(μ, θ)

The Gumbel distribution with location μ and scale θ has probability density function

$f(x; \mu, \theta) = \frac{1}{\theta} e^{-(z + e^-z)}, \quad \text{ with } z = \frac{x - \mu}{\theta}$
Gumbel()            # Gumbel distribution with zero location and unit scale, i.e. Gumbel(0, 1)
Gumbel(u)           # Gumbel distribution with location u and unit scale, i.e. Gumbel(u, 1)
Gumbel(u, b)        # Gumbel distribution with location u and scale b

params(d)        # Get the parameters, i.e. (u, b)
location(d)      # Get the location parameter, i.e. u
scale(d)         # Get the scale parameter, i.e. b

InverseGamma(α, θ)

The inverse Gamma distribution with shape parameter α and scale θ has probability density function

$f(x; \alpha, \theta) = \frac{\theta^\alpha x^{-(\alpha + 1)}}{\Gamma(\alpha)} e^{-\frac{\theta}{x}}, \quad x > 0$

It is related to the Gamma distribution: if $X \sim \operatorname{Gamma}(\alpha, \beta)$, then $1 / X \sim \operatorname{InverseGamma}(\alpha, \beta^{-1})$.

InverseGamma()        # Inverse Gamma distribution with unit shape and unit scale, i.e. InverseGamma(1, 1)
InverseGamma(α)       # Inverse Gamma distribution with shape α and unit scale, i.e. InverseGamma(α, 1)
InverseGamma(α, θ)    # Inverse Gamma distribution with shape α and scale θ

params(d)        # Get the parameters, i.e. (α, θ)
shape(d)         # Get the shape parameter, i.e. α
scale(d)         # Get the scale parameter, i.e. θ

InverseGaussian(μ,λ)

The inverse Gaussian distribution with mean μ and shape λ has probability density function

$f(x; \mu, \lambda) = \sqrt{\frac{\lambda}{2\pi x^3}} \exp\!\left(\frac{-\lambda(x-\mu)^2}{2\mu^2x}\right), \quad x > 0$
InverseGaussian()              # Inverse Gaussian distribution with unit mean and unit shape, i.e. InverseGaussian(1, 1)
InverseGaussian(mu),           # Inverse Gaussian distribution with mean mu and unit shape, i.e. InverseGaussian(u, 1)
InverseGaussian(mu, lambda)    # Inverse Gaussian distribution with mean mu and shape lambda

params(d)           # Get the parameters, i.e. (mu, lambda)
mean(d)             # Get the mean parameter, i.e. mu
shape(d)            # Get the shape parameter, i.e. lambda

Kolmogorov()

Kolmogorov distribution defined as

$\sup_{t \in [0,1]} |B(t)|$

where $B(t)$ is a Brownian bridge used in the Kolmogorov–Smirnov test for large n.

KSDist(n)

Distribution of the (two-sided) Kolmogorov-Smirnoff statistic

$D_n = \sup_x | \hat{F}_n(x) -F(x)| \sqrt(n)$

$D_n$ converges a.s. to the Kolmogorov distribution.

KSOneSided(n)

Distribution of the one-sided Kolmogorov-Smirnov test statistic:

$D^+_n = \sup_x (\hat{F}_n(x) -F(x))$
Laplace(μ,β)

The Laplace distribution with location μ and scale β has probability density function

$f(x; \mu, \beta) = \frac{1}{2 \beta} \exp \left(- \frac{|x - \mu|}{\beta} \right)$
Laplace()       # Laplace distribution with zero location and unit scale, i.e. Laplace(0, 1)
Laplace(μ)      # Laplace distribution with location μ and unit scale, i.e. Laplace(μ, 1)
Laplace(μ, β)   # Laplace distribution with location μ and scale β

params(d)       # Get the parameters, i.e., (μ, β)
location(d)     # Get the location parameter, i.e. μ
scale(d)        # Get the scale parameter, i.e. β

Levy(μ, σ)

The Lévy distribution with location μ and scale σ has probability density function

$f(x; \mu, \sigma) = \sqrt{\frac{\sigma}{2 \pi (x - \mu)^3}} \exp \left( - \frac{\sigma}{2 (x - \mu)} \right), \quad x > \mu$
Levy()         # Levy distribution with zero location and unit scale, i.e. Levy(0, 1)
Levy(u)        # Levy distribution with location u and unit scale, i.e. Levy(u, 1)
Levy(u, c)     # Levy distribution with location u ans scale c

params(d)      # Get the parameters, i.e. (u, c)
location(d)    # Get the location parameter, i.e. u

LocationScale(μ,σ,ρ)

A location-scale transformed distribution with location parameter μ, scale parameter σ, and given distribution ρ.

$f(x) = \frac{1}{σ} ρ \! \left( \frac{x-μ}{σ} \right)$
LocationScale(μ,σ,ρ) # location-scale transformed distribution
params(d)            # Get the parameters, i.e. (μ, σ, and the base distribution)
location(d)          # Get the location parameter
scale(d)             # Get the scale parameter

External links Location-Scale family on Wikipedia

Logistic(μ,θ)

The Logistic distribution with location μ and scale θ has probability density function

$f(x; \mu, \theta) = \frac{1}{4 \theta} \mathrm{sech}^2 \left( \frac{x - \mu}{2 \theta} \right)$
Logistic()       # Logistic distribution with zero location and unit scale, i.e. Logistic(0, 1)
Logistic(u)      # Logistic distribution with location u and unit scale, i.e. Logistic(u, 1)
Logistic(u, b)   # Logistic distribution with location u ans scale b

params(d)       # Get the parameters, i.e. (u, b)
location(d)     # Get the location parameter, i.e. u
scale(d)        # Get the scale parameter, i.e. b

LogitNormal(μ,σ)

The logit normal distribution is the distribution of of a random variable whose logit has a Normal distribution. Or inversely, when applying the logistic function to a Normal random variable then the resulting random variable follows a logit normal distribution.

If $X \sim \operatorname{Normal}(\mu, \sigma)$ then $\operatorname{logistic}(X) \sim \operatorname{LogitNormal}(\mu,\sigma)$.

The probability density function is

$f(x; \mu, \sigma) = \frac{1}{x \sqrt{2 \pi \sigma^2}} \exp \left( - \frac{(\text{logit}(x) - \mu)^2}{2 \sigma^2} \right), \quad x > 0$

where the logit-Function is

$\text{logit}(x) = \ln\left(\frac{x}{1-x}\right) \quad 0 < x < 1$
LogitNormal()          # Logit-normal distribution with zero logit-mean and unit scale
LogitNormal(mu)        # Logit-normal distribution with logit-mean mu and unit scale
LogitNormal(mu, sig)   # Logit-normal distribution with logit-mean mu and scale sig

params(d)            # Get the parameters, i.e. (mu, sig)
median(d)            # Get the median, i.e. logistic(mu)

The following properties have no analytical solution but numerical approximations. In order to avoid package dependencies for numerical optimization, they are currently not implemented.

mean(d)
var(d)
std(d)
mode(d)

Similarly, skewness, kurtosis, and entropy are not implemented.

LogNormal(μ,σ)

The log normal distribution is the distribution of the exponential of a Normal variate: if $X \sim \operatorname{Normal}(\mu, \sigma)$ then $\exp(X) \sim \operatorname{LogNormal}(\mu,\sigma)$. The probability density function is

$f(x; \mu, \sigma) = \frac{1}{x \sqrt{2 \pi \sigma^2}} \exp \left( - \frac{(\log(x) - \mu)^2}{2 \sigma^2} \right), \quad x > 0$
LogNormal()          # Log-normal distribution with zero log-mean and unit scale
LogNormal(mu)        # Log-normal distribution with log-mean mu and unit scale
LogNormal(mu, sig)   # Log-normal distribution with log-mean mu and scale sig

params(d)            # Get the parameters, i.e. (mu, sig)
meanlogx(d)          # Get the mean of log(X), i.e. mu
varlogx(d)           # Get the variance of log(X), i.e. sig^2
stdlogx(d)           # Get the standard deviation of log(X), i.e. sig

NoncentralBeta(α, β, λ)
NoncentralChisq(ν, λ)

The noncentral chi-squared distribution with ν degrees of freedom and noncentrality parameter λ has the probability density function

$f(x; \nu, \lambda) = \frac{1}{2} e^{-(x + \lambda)/2} \left( \frac{x}{\lambda} \right)^{\nu/4-1/2} I_{\nu/2-1}(\sqrt{\lambda x}), \quad x > 0$

It is the distribution of the sum of squares of ν independent Normal variates with individual means $\mu_i$ and

$\lambda = \sum_{i=1}^\nu \mu_i^2$
NoncentralChisq(ν, λ)     # Noncentral chi-squared distribution with ν degrees of freedom and noncentrality parameter λ

params(d)    # Get the parameters, i.e. (ν, λ)

NoncentralF(ν1, ν2, λ)
NoncentralT(ν, λ)
Normal(μ,σ)

The Normal distribution with mean μ and standard deviation σ≥0 has probability density function

$f(x; \mu, \sigma) = \frac{1}{\sqrt{2 \pi \sigma^2}} \exp \left( - \frac{(x - \mu)^2}{2 \sigma^2} \right)$

Note that if σ == 0, then the distribution is a point mass concentrated at μ. Though not technically a continuous distribution, it is allowed so as to account for cases where σ may have underflowed, and the functions are defined by taking the pointwise limit as $σ → 0$.

Normal()          # standard Normal distribution with zero mean and unit variance
Normal(mu)        # Normal distribution with mean mu and unit variance
Normal(mu, sig)   # Normal distribution with mean mu and variance sig^2

params(d)         # Get the parameters, i.e. (mu, sig)
mean(d)           # Get the mean, i.e. mu
std(d)            # Get the standard deviation, i.e. sig

NormalCanon(η, λ)

Canonical Form of Normal distribution

NormalInverseGaussian(μ,α,β,δ)

The Normal-inverse Gaussian distribution with location μ, tail heaviness α, asymmetry parameter β and scale δ has probability density function

$f(x; \mu, \alpha, \beta, \delta) = \frac{\alpha\delta K_1 \left(\alpha\sqrt{\delta^2 + (x - \mu)^2}\right)}{\pi \sqrt{\delta^2 + (x - \mu)^2}} \; e^{\delta \gamma + \beta (x - \mu)}$

where $K_j$ denotes a modified Bessel function of the third kind.

Pareto(α,θ)

The Pareto distribution with shape α and scale θ has probability density function

$f(x; \alpha, \theta) = \frac{\alpha \theta^\alpha}{x^{\alpha + 1}}, \quad x \ge \theta$
Pareto()            # Pareto distribution with unit shape and unit scale, i.e. Pareto(1, 1)
Pareto(a)           # Pareto distribution with shape a and unit scale, i.e. Pareto(a, 1)
Pareto(a, b)        # Pareto distribution with shape a and scale b

params(d)        # Get the parameters, i.e. (a, b)
shape(d)         # Get the shape parameter, i.e. a
scale(d)         # Get the scale parameter, i.e. b

PGeneralizedGaussian(α, μ, p)

The p-Generalized Gaussian distribution, more commonly known as the exponential power or the generalized normal distribution, with scale α, location μ, and shape p has the probability density function

$f(x, \mu, \alpha, p) = \frac{p}{2\alpha\Gamma(1/p)} e^{-(\frac{|x-\mu|}{\alpha})^p} \quad x \in (-\infty, +\infty) , \alpha > 0, p > 0$

The p-Generalized Gaussian (GGD) is a parametric distribution that incorporates the Normal and Laplacian distributions as special cases where p = 1 and p = 2. As p → ∞, the distribution approaches the Uniform distribution on [μ-α, μ+α].

PGeneralizedGaussian()           # GGD with shape 2, scale 1, location 0, (the Normal distribution)
PGeneralizedGaussian(loc,s,sh)   # GGD with location loc, scale s, and shape sh

params(d)                       # Get the parameters, i.e. (loc,s,sh,)
shape(d)                        # Get the shape parameter, sh
scale(d)                        # Get the scale parameter, s
location(d)                     # Get the location parameter, loc

Rayleigh(σ)

The Rayleigh distribution with scale σ has probability density function

$f(x; \sigma) = \frac{x}{\sigma^2} e^{-\frac{x^2}{2 \sigma^2}}, \quad x > 0$

It is related to the Normal distribution via the property that if $X, Y \sim \operatorname{Normal}(0,\sigma)$, independently, then $\sqrt{X^2 + Y^2} \sim \operatorname{Rayleigh}(\sigma)$.

Rayleigh()       # Rayleigh distribution with unit scale, i.e. Rayleigh(1)
Rayleigh(s)      # Rayleigh distribution with scale s

params(d)        # Get the parameters, i.e. (s,)
scale(d)         # Get the scale parameter, i.e. s

Semicircle(r)

The Wigner semicircle distribution with radius parameter r has probability density function

$f(x; r) = \frac{2}{\pi r^2} \sqrt{r^2 - x^2}, \quad x \in [-r, r].$
Semicircle(r)   # Wigner semicircle distribution with radius r

params(d)       # Get the radius parameter, i.e. (r,)

StudentizedRange(ν, k)

The studentized range distribution has probability density function:

$f(q; k, \nu) = \frac{\sqrt{2\pi}k(k - 1)\nu^{\nu/2}}{\Gamma{\left(\frac{\nu}{2}\right)}2^{\nu/2 - 1}} \int_{0}^{\infty} {x^{\nu}\phi(\sqrt{\nu}x)} \left[\int_{-\infty}^{\infty} {\phi(u)\phi(u - qx)[\Phi(u) - \Phi(u - qx)]^{k - 2}}du\right]dx$

where

\begin{align*} \Phi(x) &= \frac{1 + erf(\frac{x}{\sqrt{2}})}{2} &&(\text{Normal Distribution CDF})\\ \phi(x) &= \Phi'(x) &&(\text{Normal Distribution PDF}) \end{align*}
StudentizedRange(ν, k)     # Studentized Range Distribution with parameters ν and k

params(d)        # Get the parameters, i.e. (ν, k)

SymTriangularDist(μ,σ)

The Symmetric triangular distribution with location μ and scale σ has probability density function

$f(x; \mu, \sigma) = \frac{1}{\sigma} \left( 1 - \left| \frac{x - \mu}{\sigma} \right| \right), \quad \mu - \sigma \le x \le \mu + \sigma$
SymTriangularDist()         # Symmetric triangular distribution with zero location and unit scale
SymTriangularDist(u)        # Symmetric triangular distribution with location u and unit scale
SymTriangularDist(u, s)     # Symmetric triangular distribution with location u and scale s

params(d)       # Get the parameters, i.e. (u, s)
location(d)     # Get the location parameter, i.e. u
scale(d)        # Get the scale parameter, i.e. s
TDist(ν)

The Students T distribution with ν degrees of freedom has probability density function

$f(x; d) = \frac{1}{\sqrt{d} B(1/2, d/2)} \left( 1 + \frac{x^2}{d} \right)^{-\frac{d + 1}{2}}$
TDist(d)      # t-distribution with d degrees of freedom

params(d)     # Get the parameters, i.e. (d,)
dof(d)        # Get the degrees of freedom, i.e. d

Student's T distribution on Wikipedia

TriangularDist(a,b,c)

The triangular distribution with lower limit a, upper limit b and mode c has probability density function

$f(x; a, b, c)= \begin{cases} 0 & \mathrm{for\ } x < a, \\ \frac{2(x-a)}{(b-a)(c-a)} & \mathrm{for\ } a \le x \leq c, \\[4pt] \frac{2(b-x)}{(b-a)(b-c)} & \mathrm{for\ } c < x \le b, \\[4pt] 0 & \mathrm{for\ } b < x, \end{cases}$
TriangularDist(a, b)        # Triangular distribution with lower limit a, upper limit b, and mode (a+b)/2
TriangularDist(a, b, c)     # Triangular distribution with lower limit a, upper limit b, and mode c

params(d)       # Get the parameters, i.e. (a, b, c)
minimum(d)      # Get the lower bound, i.e. a
maximum(d)      # Get the upper bound, i.e. b
mode(d)         # Get the mode, i.e. c

Triweight(μ, σ)
Uniform(a,b)

The continuous uniform distribution over an interval $[a, b]$ has probability density function

$f(x; a, b) = \frac{1}{b - a}, \quad a \le x \le b$
Uniform()        # Uniform distribution over [0, 1]
Uniform(a, b)    # Uniform distribution over [a, b]

params(d)        # Get the parameters, i.e. (a, b)
minimum(d)       # Get the lower bound, i.e. a
maximum(d)       # Get the upper bound, i.e. b
location(d)      # Get the location parameter, i.e. a
scale(d)         # Get the scale parameter, i.e. b - a

VonMises(μ, κ)

The von Mises distribution with mean μ and concentration κ has probability density function

$f(x; \mu, \kappa) = \frac{1}{2 \pi I_0(\kappa)} \exp \left( \kappa \cos (x - \mu) \right)$
VonMises()       # von Mises distribution with zero mean and unit concentration
VonMises(κ)      # von Mises distribution with zero mean and concentration κ
VonMises(μ, κ)   # von Mises distribution with mean μ and concentration κ

Weibull(α,θ)

The Weibull distribution with shape α and scale θ has probability density function

$f(x; \alpha, \theta) = \frac{\alpha}{\theta} \left( \frac{x}{\theta} \right)^{\alpha-1} e^{-(x/\theta)^\alpha}, \quad x \ge 0$
Weibull()        # Weibull distribution with unit shape and unit scale, i.e. Weibull(1, 1)
Weibull(a)       # Weibull distribution with shape a and unit scale, i.e. Weibull(a, 1)
Weibull(a, b)    # Weibull distribution with shape a and scale b

params(d)        # Get the parameters, i.e. (a, b)
shape(d)         # Get the shape parameter, i.e. a
scale(d)         # Get the scale parameter, i.e. b

## Discrete Distributions

Bernoulli(p)

A Bernoulli distribution is parameterized by a success rate p, which takes value 1 with probability p and 0 with probability 1-p.

$P(X = k) = \begin{cases} 1 - p & \quad \text{for } k = 0, \\ p & \quad \text{for } k = 1. \end{cases}$
Bernoulli()    # Bernoulli distribution with p = 0.5
Bernoulli(p)   # Bernoulli distribution with success rate p

params(d)      # Get the parameters, i.e. (p,)
succprob(d)    # Get the success rate, i.e. p
failprob(d)    # Get the failure rate, i.e. 1 - p

BetaBinomial(n,α,β)

A Beta-binomial distribution is the compound distribution of the Binomial distribution where the probability of success p is distributed according to the Beta. It has three parameters: n, the number of trials and two shape parameters α, β

$P(X = k) = {n \choose k} B(k + \alpha, n - k + \beta) / B(\alpha, \beta), \quad \text{ for } k = 0,1,2, \ldots, n.$
BetaBinomial(n, a, b)      # BetaBinomial distribution with n trials and shape parameters a, b

params(d)       # Get the parameters, i.e. (n, a, b)
ntrials(d)      # Get the number of trials, i.e. n

Binomial(n,p)

A Binomial distribution characterizes the number of successes in a sequence of independent trials. It has two parameters: n, the number of trials, and p, the probability of success in an individual trial, with the distribution:

$P(X = k) = {n \choose k}p^k(1-p)^{n-k}, \quad \text{ for } k = 0,1,2, \ldots, n.$
Binomial()      # Binomial distribution with n = 1 and p = 0.5
Binomial(n)     # Binomial distribution for n trials with success rate p = 0.5
Binomial(n, p)  # Binomial distribution for n trials with success rate p

params(d)       # Get the parameters, i.e. (n, p)
ntrials(d)      # Get the number of trials, i.e. n
succprob(d)     # Get the success rate, i.e. p
failprob(d)     # Get the failure rate, i.e. 1 - p

Categorical(p)

A Categorical distribution is parameterized by a probability vector p (of length K).

$P(X = k) = p[k] \quad \text{for } k = 1, 2, \ldots, K.$
Categorical(p)   # Categorical distribution with probability vector p
params(d)        # Get the parameters, i.e. (p,)
probs(d)         # Get the probability vector, i.e. p
ncategories(d)   # Get the number of categories, i.e. K

Here, p must be a real vector, of which all components are nonnegative and sum to one.

Note: The input vector p is directly used as a field of the constructed distribution, without being copied.

Categorical is simply a type alias describing a special case of a DiscreteNonParametric distribution, so non-specialized methods defined for DiscreteNonParametric apply to Categorical as well.

DiscreteUniform(a,b)

A Discrete uniform distribution is a uniform distribution over a consecutive sequence of integers between a and b, inclusive.

$P(X = k) = 1 / (b - a + 1) \quad \text{for } k = a, a+1, \ldots, b.$
DiscreteUniform(a, b)   # a uniform distribution over {a, a+1, ..., b}

params(d)       # Get the parameters, i.e. (a, b)
span(d)         # Get the span of the support, i.e. (b - a + 1)
probval(d)      # Get the probability value, i.e. 1 / (b - a + 1)
minimum(d)      # Return a
maximum(d)      # Return b

DiscreteNonParametric(xs, ps)

A Discrete nonparametric distribution explicitly defines an arbitrary probability mass function in terms of a list of real support values and their corresponding probabilities

d = DiscreteNonParametric(xs, ps)

params(d)  # Get the parameters, i.e. (xs, ps)
support(d) # Get a sorted AbstractVector describing the support (xs) of the distribution
probs(d)   # Get a Vector of the probabilities (ps) associated with the support

Geometric(p)

A Geometric distribution characterizes the number of failures before the first success in a sequence of independent Bernoulli trials with success rate p.

$P(X = k) = p (1 - p)^k, \quad \text{for } k = 0, 1, 2, \ldots.$
Geometric()    # Geometric distribution with success rate 0.5
Geometric(p)   # Geometric distribution with success rate p

params(d)      # Get the parameters, i.e. (p,)
succprob(d)    # Get the success rate, i.e. p
failprob(d)    # Get the failure rate, i.e. 1 - p

Hypergeometric(s, f, n)

A Hypergeometric distribution describes the number of successes in n draws without replacement from a finite population containing s successes and f failures.

$P(X = k) = {{{s \choose k} {f \choose {n-k}}}\over {s+f \choose n}}, \quad \text{for } k = \max(0, n - f), \ldots, \min(n, s).$
Hypergeometric(s, f, n)  # Hypergeometric distribution for a population with
# s successes and f failures, and a sequence of n trials.

params(d)       # Get the parameters, i.e. (s, f, n)

NegativeBinomial(r,p)

A Negative binomial distribution describes the number of failures before the rth success in a sequence of independent Bernoulli trials. It is parameterized by r, the number of successes, and p, the probability of success in an individual trial.

$P(X = k) = {k + r - 1 \choose k} p^r (1 - p)^k, \quad \text{for } k = 0,1,2,\ldots.$

The distribution remains well-defined for any positive r, in which case

$P(X = k) = \frac{\Gamma(k+r)}{k! \Gamma(r)} p^r (1 - p)^k, \quad \text{for } k = 0,1,2,\ldots.$
NegativeBinomial()        # Negative binomial distribution with r = 1 and p = 0.5
NegativeBinomial(r, p)    # Negative binomial distribution with r successes and success rate p

params(d)       # Get the parameters, i.e. (r, p)
succprob(d)     # Get the success rate, i.e. p
failprob(d)     # Get the failure rate, i.e. 1 - p

Note: The definition of the negative binomial distribution in Wolfram is different from the Wikipedia definition. In Wikipedia, r is the number of failures and k is the number of successes.

Poisson(λ)

A Poisson distribution descibes the number of independent events occurring within a unit time interval, given the average rate of occurrence λ.

$P(X = k) = \frac{\lambda^k}{k!} e^{-\lambda}, \quad \text{ for } k = 0,1,2,\ldots.$
Poisson()        # Poisson distribution with rate parameter 1
Poisson(lambda)       # Poisson distribution with rate parameter lambda

params(d)        # Get the parameters, i.e. (λ,)
mean(d)          # Get the mean arrival rate, i.e. λ

PoissonBinomial(p)

A Poisson-binomial distribution describes the number of successes in a sequence of independent trials, wherein each trial has a different success rate. It is parameterized by a vector p (of length $K$), where $K$ is the total number of trials and p[i] corresponds to the probability of success of the ith trial.

$P(X = k) = \sum\limits_{A\in F_k} \prod\limits_{i\in A} p[i] \prod\limits_{j\in A^c} (1-p[j]), \quad \text{ for } k = 0,1,2,\ldots,K,$

where $F_k$ is the set of all subsets of $k$ integers that can be selected from $\{1,2,3,...,K\}$.

PoissonBinomial(p)   # Poisson Binomial distribution with success rate vector p

params(d)            # Get the parameters, i.e. (p,)
succprob(d)          # Get the vector of success rates, i.e. p
failprob(d)          # Get the vector of failure rates, i.e. 1-p

Skellam(μ1, μ2)

A Skellam distribution describes the difference between two independent Poisson variables, respectively with rate μ1 and μ2.

$P(X = k) = e^{-(\mu_1 + \mu_2)} \left( \frac{\mu_1}{\mu_2} \right)^{k/2} I_k(2 \sqrt{\mu_1 \mu_2}) \quad \text{for integer } k$

where $I_k$ is the modified Bessel function of the first kind.

Skellam(mu1, mu2)   # Skellam distribution for the difference between two Poisson variables,
# respectively with expected values mu1 and mu2.

params(d)           # Get the parameters, i.e. (mu1, mu2)