List of 1D Continuous Distributions

The following is a list of all 1D continuous distributions currently available in UQpy.

Beta

Beta distribution having probability density function

\[f(x|a,b) = \dfrac{ \Gamma(a+b)x^{a-1}(1-x)^{b-1}}{\Gamma(a) \Gamma(b)}\]

for \(0 \le x \ge 0\), \(a > 0, b > 0\). Here \(\Gamma (a)\) refers to the Gamma function.

In this standard form \((loc=0, scale=1)\), the distribution is defined over the interval (0, 1). Use loc and scale to shift the distribution to interval \((loc, loc + scale)\). Specifically, this is equivalent to computing \(f(y|a,b)\) where \(y=(x-loc)/scale\).

The Beta class is imported using the following command:

>>> from UQpy.distributions.collection.Beta import Beta
class Beta(a, b, loc=0.0, scale=1.0)[source]
Parameters:

Cauchy

Cauchy distribution having probability density function

\[f(x) = \dfrac{1}{\pi(1+x^2)}\]

In this standard form \((loc=0, scale=1)\). Use loc and scale to shift and scale the distribution. Specifically, this is equivalent to computing \(f(y)\) where \(y=(x-loc)/scale\).

The Cauchy class is imported using the following command:

>>> from UQpy.distributions.collection.Cauchy import Cauchy
class Cauchy(loc=0.0, scale=1.0)[source]
Parameters:

Chi Square

Chi-square distribution having probability density:

\[f(x|k) = \dfrac{1}{2^{k/2}\Gamma(k/2)}x^{k/2-1}\exp{(-x/2)}\]

for \(x\ge 0\), \(k>0\). Here \(\Gamma(\cdot)\) refers to the Gamma function.

In this standard form \((loc=0, scale=1)\). Use loc and scale to shift and scale the distribution. Specifically, this is equivalent to computing \(f(y|k)\) where \(y=(x-loc)/scale\).

The ChiSquare class is imported using the following command:

>>> from UQpy.distributions.collection.ChiSquare import ChiSquare
class ChiSquare(df, loc=0.0, scale=1.0)[source]
Parameters:

Exponential

Exponential distribution having probability density function:

\[f(x) = \exp(-x)\]

In this standard form \((loc=0, scale=1)\). Use loc and scale to shift and scale the distribution. Specifically, this is equivalent to computing \(f(y)\) where \(y=(x-loc)/scale\).

A common parameterization for Exponential is in terms of the rate parameter \(\lambda\), which corresponds to using \(scale = 1 / \lambda\).

The Exponential class is imported using the following command:

>>> from UQpy.distributions.collection.ExponentialCorrelation import Exponential

>>> from UQpy.distributions.collection.ExponentialCorrelation import Exponential

>>> from UQpy.distributions.collection.Exponential import Exponential
class Exponential(loc=0.0, scale=1.0)[source]
Parameters:

Gamma

Gamma distribution having probability density function:

\[f(x|a) = \dfrac{x^{a-1}\exp(-x)}{\Gamma(a)}\]

for \(x\ge 0\), \(a>0\). Here \(\Gamma(a)\) refers to the Gamma function.

In this standard form \((loc=0, scale=1)\). Use loc and scale to shift and scale the distribution. Specifically, this is equivalent to computing \(f(y)\) where \(y=(x-loc)/scale\).

The Gamma class is imported using the following command:

>>> from UQpy.distributions.collection.Gamma import Gamma
class Gamma(a, loc=0.0, scale=1.0)[source]
Parameters:

Generalized Extreme

Generalized Extreme Value distribution having probability density function:

\[f(x|c) = \exp(-(1-cx)^{1/c})(1-cx)^{1/c-1}\]

for \(x\le 1/c, c>0\).

For \(c=0\)

\[f(x) = \exp(\exp(-x))\exp(-x)\]

In this standard form \((loc=0, scale=1)\). Use loc and scale to shift and scale the distribution. Specifically, this is equivalent to computing \(f(y)\) where \(y=(x-loc)/scale\).

The GeneralizedExtreme class is imported using the following command:

>>> from UQpy.distributions.collection.GeneralizedExtreme import GeneralizedExtreme
class GeneralizedExtreme(c, loc=0.0, scale=1.0)[source]
Parameters:

Inverse Gaussian

Inverse Gaussian distribution having probability density function

\[\begin{split}f(x|\mu) = \dfrac{1}{2\pi x^3}\exp{(-\dfrac{(x\\mu)^2}{2x\mu^2})}\end{split}\]

for \(x>0\). cdf() method returns NaN for \(\mu<0.0028\).

In this standard form \((loc=0, scale=1)\). Use loc and scale to shift and scale the distribution. Specifically, this is equivalent to computing \(f(y)\) where \(y=(x-loc)/scale\).

The InverseGauss class is imported using the following command:

>>> from UQpy.distributions.collection.InverseGaussian import InverseGauss
class InverseGauss(mu, loc=0.0, scale=1.0)[source]
Parameters:

Laplace

Laplace distribution having probability density function

\[f(x) = \dfrac{1}{2}\exp{-|x|}\]

In this standard form \((loc=0, scale=1)\). Use loc and scale to shift and scale the distribution. Specifically, this is equivalent to computing \(f(y)\) where \(y=(x-loc)/scale\).

The Laplace class is imported using the following command:

>>> from UQpy.distributions.collection.Laplace import Laplace
class Laplace(loc=0.0, scale=1.0)[source]
Parameters:

Levy

Levy distribution having probability density function

\[f(x) = \dfrac{1}{\sqrt{2\pi x^3}}\exp(-\dfrac{1}{2x})\]

for \(x\ge 0\).

In this standard form \((loc=0, scale=1)\). Use loc and scale to shift and scale the distribution. Specifically, this is equivalent to computing \(f(y)\) where \(y=(x-loc)/scale\).

The Levy class is imported using the following command:

>>> from UQpy.distributions.collection.Levy import Levy
class Levy(loc=0.0, scale=1.0)[source]
Parameters:

Logistic

Logistic distribution having probability density function

\[f(x) = \dfrac{\exp(-x)}{(1+\exp(-x))^2}\]

In this standard form \((loc=0, scale=1)\). Use loc and scale to shift and scale the distribution. Specifically, this is equivalent to computing \(f(y)\) where \(y=(x-loc)/scale\).

The Logistic class is imported using the following command:

>>> from UQpy.distributions.collection.Logistic import Logistic
class Logistic(loc=0.0, scale=1.0)[source]
Parameters:

Lognormal

Lognormal distribution having probability density function

\[f(x|s) = \dfrac{1}{sx\sqrt{2\pi}}\exp(-\dfrac{\log^2(x)}{2s^2})\]

for \(x>0, s>0\).

A common parametrization for a lognormal random variable \(Y\) is in terms of the mean, mu, and standard deviation, sigma, of the gaussian random variable \(X\) such that \(exp(X) = Y\). This parametrization corresponds to setting \(s = sigma\) and \(scale = exp(mu)\).

The Lognormal class is imported using the following command:

>>> from UQpy.distributions.collection.Lognormal import Lognormal
class Lognormal(s, loc=0.0, scale=1.0)[source]
Parameters:

Maxwell

Maxwell-Boltzmann distribution having probability density function

\[f(x) = \sqrt{2/\pi}x^2\exp(-x^2/2)\]

for \(x\ge0\).

In this standard form \((loc=0, scale=1)\). Use loc and scale to shift and scale the distribution. Specifically, this is equivalent to computing \(f(y)\) where \(y=(x-loc)/scale\).

The Maxwell class is imported using the following command:

>>> from UQpy.distributions.collection.Maxwell import Maxwell
class Maxwell(loc=0.0, scale=1.0)[source]
Parameters:

Normal

Normal distribution having probability density function

\[f(x) = \dfrac{\exp(-x^2/2)}{\sqrt{2\pi}}\]

In this standard form \((loc=0, scale=1)\). Use loc and scale to shift and scale the distribution. Specifically, this is equivalent to computing \(f(y)\) where \(y=(x-loc)/scale\).

The Normal class is imported using the following command:

>>> from UQpy.distributions.collection.Normal import Normal
class Normal(loc=0.0, scale=1.0)[source]
Parameters:

Pareto

Pareto distribution having probability density function

\[f(x|b) = \dfrac{b}{x^{b+1}}\]

for \(x\ge 1, b>0\).

In this standard form \((loc=0, scale=1)\). Use loc and scale to shift and scale the distribution. Specifically, this is equivalent to computing \(f(y)\) where \(y=(x-loc)/scale\).

The Pareto class is imported using the following command:

>>> from UQpy.distributions.collection.Pareto import Pareto
class Pareto(b, loc=0.0, scale=1.0)[source]
Parameters:

Rayleigh

Rayleigh distribution having probability density function

\[f(x) = x\exp(-x^2/2)\]

for \(x\ge 0\).

In this standard form \((loc=0, scale=1)\). Use loc and scale to shift and scale the distribution. Specifically, this is equivalent to computing \(f(y)\) where \(y=(x-loc)/scale\).

The Rayleigh class is imported using the following command:

>>> from UQpy.distributions.collection.Rayleigh import Rayleigh
class Rayleigh(loc=0.0, scale=1.0)[source]
Parameters:

Truncated Normal

Truncated normal distribution

The standard form of this distribution \((loc=0, scale=1)\) is a standard normal truncated to the range \([a, b]\). Note that a and b are defined over the domain of the standard normal.

The TruncatedNormal class is imported using the following command:

>>> from UQpy.distributions.collection.TruncatedNormal import TruncatedNormal
class TruncatedNormal(a, b, loc=0.0, scale=1.0)[source]
Parameters:

Uniform

Uniform distribution having probability density function

\[f(x|a, b) = \dfrac{1}{b-a}\]

where \(a=loc\) and \(b=loc+scale\)

The Uniform class is imported using the following command:

>>> from UQpy.distributions.collection.Uniform import Uniform
class Uniform(loc=0.0, scale=1.0)[source]
Parameters: