# Univariate 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


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(1.0, 1.0)
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(0.0, 1.0)
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)


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(0.0, 1.0)
Cauchy(u)        # Cauchy distribution with location u and unit scale, i.e. Cauchy(u, 1.0)
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


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


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(1.0, 1.0)
Erlang(a)      # Erlang distribution with shape parameter a and unit scale, i.e. Erlang(a, 1.0)
Erlang(a, s)   # Erlang distribution with shape parameter a and scale b


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(1.0)
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 FDist(\nu_1, \nu_2).

FDist(d1, d2)     # F-Distribution with parameters d1 and d2

params(d)         # Get the parameters, i.e. (d1, d2)


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(1.0, 1.0)
Frechet(a)       # Fréchet distribution with shape a and unit scale, i.e. Frechet(a, 1.0)
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(1.0, 1.0)
Gamma(a)         # Gamma distribution with shape a and unit scale, i.e. Gamma(a, 1.0)
Gamma(a, b)      # Gamma 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


GeneralizedExtremeValue(μ, σ, ξ)

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

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

for

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

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


GeneralizedPareto(ξ, σ, μ)

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

$\begin{split}f(x; \xi, \sigma, \mu) = \begin{cases} \frac{1}{\sigma}(1 + \xi \frac{x - \mu}{\sigma} )^{-\frac{1}{\xi} - 1} & \text{for } \xi \neq 0 \\ \frac{1}{\sigma} e^{-\frac{\left( x - \mu \right) }{\sigma}} & \text{for } \xi = 0 \end{cases}~, \quad x \in \begin{cases} \left[ \mu, \infty \right] & \text{for } \xi \geq 0 \\ \left[ \mu, \mu - \sigma / \xi \right] & \text{for } \xi < 0 \end{cases}\end{split}$
GeneralizedPareto()             # Generalized Pareto distribution with unit shape and unit scale, i.e. GeneralizedPareto(1.0, 1.0, 0.0)
GeneralizedPareto(k, s)         # Generalized Pareto distribution with shape k and scale s, i.e. GeneralizedPareto(k, s, 0.0)
GeneralizedPareto(k, s, m)      # Generalized Pareto distribution with shape k, scale s and location m.

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


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(0.0, 1.0)
Gumbel(u)           # Gumbel distribution with location u and unit scale, i.e. Gumbel(u, 1.0)
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(1.0, 1.0)
InverseGamma(a)       # Inverse Gamma distribution with shape a and unit scale, i.e. InverseGamma(a, 1.0)
InverseGamma(a, b)    # Inverse Gamma 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


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(1.0, 1.0)
InverseGaussian(mu),           # Inverse Gaussian distribution with mean mu and unit shape, i.e. InverseGaussian(u, 1.0)
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


Laplace(μ, θ)

The Laplace distribution with location μ and scale θ 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(0.0, 1.0)
Laplace(u)      # Laplace distribution with location u and unit scale, i.e. Laplace(u, 1.0)
Laplace(u, b)   # Laplace 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


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(0.0, 1.0)
Levy(u)        # Levy distribution with location u and unit scale, i.e. Levy(u, 1.0)
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


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


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(0.0, 1.0)
Logistic(u)      # Logistic distribution with location u and unit scale, i.e. Logistic(u, 1.0)
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


Normal(μ, σ)

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

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


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(1.0, 1.0)
Pareto(a)           # Pareto distribution with shape a and unit scale, i.e. Pareto(a, 1.0)
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


External links * Pareto distribution on Wikipedia

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(1.0)
Rayleigh(s)      # Rayleigh distribution with scale s

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


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

$\begin{split}f(x; a, b, c)= \begin{cases} 0 & \mathrm{for\ } x < a, \\ \frac{2(x-a)}{(b-a)(c-a)} & \mathrm{for\ } a \le x \leq c, \\[4pt] \frac{2(b-x)}{(b-a)(b-c)} & \mathrm{for\ } c < x \le b, \\[4pt] 0 & \mathrm{for\ } b < x, \end{cases}\end{split}$
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


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(1.0, 1.0)
Weibull(a)       # Weibull distribution with shape a and unit scale, i.e. Weibull(a, 1.0)
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
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