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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 is implemented for each univariate distribution, which provides 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.

StatsAPI.params — Method
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 $d$.

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

Computation of statistics

Base.maximum — Method
maximum(d::Distribution)

Return the maximum of the support of d.

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Base.minimum — Method
minimum(d::Distribution)

Return the minimum of the support of d.

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Base.extrema — Method
extrema(d::Distribution)

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

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Statistics.var — Method
var(d::UnivariateDistribution)

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

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Statistics.std — Method
std(d::UnivariateDistribution)

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

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Statistics.median — Method
median(d::UnivariateDistribution)

Return the median value of distribution d. The median is the smallest x in the support of d for which cdf(d, x) ≥ 1/2. Corresponding to this definition as 1/2-quantile, a fallback is provided calling the quantile function.

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StatsBase.kurtosis — Method
kurtosis(d::Distribution, correction::Bool)

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

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StatsBase.entropy — Method
entropy(d::UnivariateDistribution, b::Real)

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

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StatsBase.entropy — Method
entropy(d::UnivariateDistribution, b::Real)

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

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Distributions.cf — Method
cf(d::UnivariateDistribution, t)

Evaluate the characteristic function of distribution d.

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Distributions.pdfsquaredL2norm — Function
pdfsquaredL2norm(d::Distribution)

Return the square of the L2 norm of the probability density function $f(x)$ of the distribution d:

\[\int_{S} f(x)^{2} \mathrm{d} x\]

where $S$ is the support of $f(x)$.

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Probability Evaluation

Distributions.insupport — Method
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.

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Distributions.logpdf — Method
logpdf(d::UnivariateDistribution, x::Real)

Evaluate the logarithm of probability density (mass) at x.

See also: pdf.

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Missing docstring.

Missing docstring for loglikelihood(::UnivariateDistribution, ::AbstractArray). Check Documenter's build log for details.

Distributions.logcdf — Method
logcdf(d::UnivariateDistribution, x::Real)

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

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Distributions.logdiffcdf — Method
logdiffcdf(d::UnivariateDistribution, x::Real, y::Real)

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

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Distributions.ccdf — Method
ccdf(d::UnivariateDistribution, x::Real)

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

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Distributions.logccdf — Method
logccdf(d::UnivariateDistribution, x::Real)

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

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Statistics.quantile — Method
quantile(d::UnivariateDistribution, q::Real)

Evaluate the (generalized) inverse cumulative distribution function at q.

For a given 0 ≤ q ≤ 1, quantile(d, q) is the smallest value x in the support of d for which cdf(d, x) ≥ q.

See also: cquantile, invlogcdf, and invlogccdf.

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Distributions.invlogcdf — Method
invlogcdf(d::UnivariateDistribution, lp::Real)

The (generalized) inverse function of logcdf.

For a given lp ≤ 0, invlogcdf(d, lp) is the smallest value x in the support of d for which logcdf(d, x) ≥ lp.

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Distributions.invlogccdf — Method
invlogccdf(d::UnivariateDistribution, lp::Real)

The (generalized) inverse function of logccdf.

For a given lp ≤ 0, invlogccdf(d, lp) is the smallest value x in the support of d for which logccdf(d, x) ≤ lp.

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Sampling (Random number generation)

Base.rand — Method
rand(rng::AbstractRNG, d::UnivariateDistribution)

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

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Random.rand! — Method
rand!(::AbstractRNG, ::Sampleable, ::AbstractArray)

Samples in-place from the sampler and stores the result in the provided array.

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Continuous Distributions

Distributions.Arcsine — Type
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

External links

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

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Distributions.Beta — Type
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 $X / (X + Y) \sim \operatorname{Beta}(\alpha, \beta)$.

Beta()        # equivalent to Beta(1, 1)
Beta(α)       # equivalent to Beta(α, α)
Beta(α, β)    # Beta distribution with shape parameters α and β

params(d)     # Get the parameters, i.e. (α, β)

External links

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Distributions.BetaPrime — Type
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 relationship 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(α)       # equivalent to BetaPrime(α, α)
BetaPrime(α, β)    # Beta prime distribution with shape parameters α and β

params(d)          # Get the parameters, i.e. (α, β)

External links

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Distributions.Cauchy — Type
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(μ)        # Cauchy distribution with location μ and unit scale, i.e. Cauchy(μ, 1)
Cauchy(μ, σ)     # Cauchy distribution with location μ and scale σ

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

External links

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Distributions.Chernoff — Type
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.

cdf(Chernoff(),-x)              # For tail probabilities, use this instead of 1-cdf(Chernoff(),x)
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Distributions.Chi — Type
Chi(ν)

The Chi distribution ν degrees of freedom has probability density function

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

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

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

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

External links

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Distributions.Chisq — Type
Chisq(ν)

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

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

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

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

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

External links

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Distributions.Erlang — Type
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

External links

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Distributions.Exponential — Type
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(θ)     # Exponential distribution with scale θ

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

External links

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Distributions.FDist — Type
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)

External links

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Distributions.Frechet — Type
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(α)       # Fréchet distribution with shape α and unit scale, i.e. Frechet(α, 1)
Frechet(α, θ)    # Fréchet 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

External links

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Distributions.Gamma — Type
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

External links

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Distributions.GeneralizedExtremeValue — Type
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(μ, σ, ξ)      # Generalized Pareto distribution with shape ξ, scale σ and location μ.

params(d)       # Get the parameters, i.e. (μ, σ, ξ)
location(d)     # Get the location parameter, i.e. μ
scale(d)        # Get the scale parameter, i.e. σ
shape(d)        # Get the shape parameter, i.e. ξ (sometimes called c)

External links

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Distributions.GeneralizedPareto — Type
GeneralizedPareto(μ, σ, ξ)

The Generalized Pareto distribution (GPD) 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()             # GPD with unit shape and unit scale, i.e. GeneralizedPareto(0, 1, 1)
GeneralizedPareto(ξ)            # GPD with shape ξ and unit scale, i.e. GeneralizedPareto(0, 1, ξ)
GeneralizedPareto(σ, ξ)         # GPD with shape ξ and scale σ, i.e. GeneralizedPareto(0, σ, ξ)
GeneralizedPareto(μ, σ, ξ)      # GPD with shape ξ, scale σ and location μ.

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

External links

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Distributions.Gumbel — Type
Gumbel(μ, θ)

The Gumbel (maxima) 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(μ)           # Gumbel distribution with location μ and unit scale, i.e. Gumbel(μ, 1)
Gumbel(μ, θ)        # Gumbel distribution with location μ and scale θ

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

External links

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Distributions.InverseGamma — Type
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

External links

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Distributions.InverseGaussian — Type
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(μ),            # Inverse Gaussian distribution with mean μ and unit shape, i.e. InverseGaussian(μ, 1)
InverseGaussian(μ, λ)          # Inverse Gaussian distribution with mean μ and shape λ

params(d)           # Get the parameters, i.e. (μ, λ)
mean(d)             # Get the mean parameter, i.e. μ
shape(d)            # Get the shape parameter, i.e. λ

External links

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Distributions.JohnsonSU — Type
JohnsonSU(ξ, λ, γ, δ)

The Johnson's $S_U$-distribution with parameters ξ, λ, γ and δ is a transformation of the normal distribution:

If

\[X = \lambda\sinh\Bigg( \frac{Z - \gamma}{\delta} \Bigg) + \xi\]

where $Z \sim \mathcal{N}(0,1)$, then $X \sim \operatorname{Johnson}(\xi, \lambda, \gamma, \delta)$.

JohnsonSU()           # Equivalent to JohnsonSU(0, 1, 0, 1)
JohnsonSU(ξ, λ, γ, δ) # JohnsonSU's S_U-distribution with parameters ξ, λ, γ and δ

params(d)           # Get the parameters, i.e. (ξ, λ, γ, δ)
shape(d)            # Get the shape parameter, i.e. ξ
scale(d)            # Get the scale parameter, i.e. λ

External links

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Distributions.Kolmogorov — Type
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.

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Distributions.KSDist — Type
KSDist(n)

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

\[D_n = \sup_x | \hat{F}_n(x) -F(x)|\]

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

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Distributions.KSOneSided — Type
KSOneSided(n)

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

\[D^+_n = \sup_x (\hat{F}_n(x) -F(x))\]

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Distributions.Kumaraswamy — Type
Kumaraswamy(a, b)

The Kumaraswamy distribution with shape parameters a > 0 and b > 0 has probability density function

\[f(x; a, b) = a b x^{a - 1} (1 - x^a)^{b - 1}, \quad 0 < x < 1\]

It is related to the Beta distribution by the following identity: if $X \sim \operatorname{Kumaraswamy}(a, b)$ then $X^a \sim \operatorname{Beta}(1, b)$. In particular, if $X \sim \operatorname{Kumaraswamy}(1, 1)$ then $X \sim \operatorname{Uniform}(0, 1)$.

External links

References

  • Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes. Journal of Hydrology. 46(1-2), 79-88.
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Distributions.Laplace — Type
Laplace(μ,θ)

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

\[f(x; \mu, \theta) = \frac{1}{2 \theta} \exp \left(- \frac{|x - \mu|}{\theta} \right)\]

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, θ)   # Laplace distribution with location μ and scale θ

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

External links

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Distributions.Levy — Type
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(μ)        # Levy distribution with location μ and unit scale, i.e. Levy(μ, 1)
Levy(μ, σ)     # Levy distribution with location μ and scale σ

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

External links

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Distributions.Lindley — Type
Lindley(θ)

The one-parameter Lindley distribution with shape θ > 0 has probability density function

\[f(x; \theta) = \frac{\theta^2}{1 + \theta} (1 + x) e^{-\theta x}, \quad x > 0\]

It was first described by Lindley[1] and was studied in greater detail by Ghitany et al.[2] Note that Lindley(θ) is a mixture of an Exponential(θ) and a Gamma(2, θ) with respective mixing weights p = θ/(1 + θ) and 1 - p.

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Distributions.Logistic — Type
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(μ)      # Logistic distribution with location μ and unit scale, i.e. Logistic(μ, 1)
Logistic(μ, θ)   # Logistic distribution with location μ and scale θ

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

External links

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Distributions.LogitNormal — Type
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(μ)       # Logit-normal distribution with logit-mean μ and unit scale
LogitNormal(μ, σ)    # Logit-normal distribution with logit-mean μ and scale σ

params(d)            # Get the parameters, i.e. (μ, σ)
median(d)            # Get the median, i.e. logistic(μ)

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.

External links

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Distributions.LogNormal — Type
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(μ)         # Log-normal distribution with log-mean mu and unit scale
LogNormal(μ, σ)      # Log-normal distribution with log-mean mu and scale sig

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

External links

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Distributions.NoncentralBeta — Type
NoncentralBeta(α, β, λ)

Noncentral Beta distribution with shape parameters α > 0 and β > 0 and noncentrality parameter λ >= 0.

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Distributions.NoncentralChisq — Type
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)

External links

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Distributions.NoncentralF — Type
NoncentralF(ν1, ν2, λ)

Noncentral F-distribution with ν1 > 0 and ν2 > 0 degrees of freedom and noncentrality parameter λ >= 0.

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Distributions.NoncentralT — Type
NoncentralT(ν, λ)

Noncentral Student's t-distribution with v > 0 degrees of freedom and noncentrality parameter λ.

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