The following is a list of all 1D continuous distributions currently available in :py:mod:`UQpy`.
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 :class:`.Beta` class is imported using the following command:
>>> from UQpy.distributions.collection.Beta import Beta.. autoclass:: UQpy.distributions.collection.Beta
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 :class:`.Cauchy` class is imported using the following command:
>>> from UQpy.distributions.collection.Cauchy import Cauchy.. autoclass:: UQpy.distributions.collection.Cauchy
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 :class:`.ChiSquare` class is imported using the following command:
>>> from UQpy.distributions.collection.ChiSquare import ChiSquare.. autoclass:: UQpy.distributions.collection.ChiSquare
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 :class:`.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.. autoclass:: UQpy.distributions.collection.Exponential
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 :class:`.Gamma` class is imported using the following command:
>>> from UQpy.distributions.collection.Gamma import Gamma.. autoclass:: UQpy.distributions.collection.Gamma
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 :class:`.GeneralizedExtreme` class is imported using the following command:
>>> from UQpy.distributions.collection.GeneralizedExtreme import GeneralizedExtreme.. autoclass:: UQpy.distributions.collection.GeneralizedExtreme
Inverse Gaussian distribution having probability density function
f(x|\mu) = \dfrac{1}{2\pi x^3}\exp{(-\dfrac{(x\\mu)^2}{2x\mu^2})}
for x>0. :py:meth:`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 :class:`.InverseGauss` class is imported using the following command:
>>> from UQpy.distributions.collection.InverseGaussian import InverseGauss.. autoclass:: UQpy.distributions.collection.InverseGauss
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 :class:`.Laplace` class is imported using the following command:
>>> from UQpy.distributions.collection.Laplace import Laplace.. autoclass:: UQpy.distributions.collection.Laplace
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 :class:`.Levy` class is imported using the following command:
>>> from UQpy.distributions.collection.Levy import Levy.. autoclass:: UQpy.distributions.collection.Levy
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 :class:`.Logistic` class is imported using the following command:
>>> from UQpy.distributions.collection.Logistic import Logistic.. autoclass:: UQpy.distributions.collection.Logistic
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 :class:`.Lognormal` class is imported using the following command:
>>> from UQpy.distributions.collection.Lognormal import Lognormal.. autoclass:: UQpy.distributions.collection.Lognormal
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 :class:`.Maxwell` class is imported using the following command:
>>> from UQpy.distributions.collection.Maxwell import Maxwell.. autoclass:: UQpy.distributions.collection.Maxwell
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 :class:`.Normal` class is imported using the following command:
>>> from UQpy.distributions.collection.Normal import Normal.. autoclass:: UQpy.distributions.collection.Normal
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 :class:`.Pareto` class is imported using the following command:
>>> from UQpy.distributions.collection.Pareto import Pareto.. autoclass:: UQpy.distributions.collection.Pareto
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 :class:`.Rayleigh` class is imported using the following command:
>>> from UQpy.distributions.collection.Rayleigh import Rayleigh.. autoclass:: UQpy.distributions.collection.Rayleigh
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 :class:`.TruncatedNormal` class is imported using the following command:
>>> from UQpy.distributions.collection.TruncatedNormal import TruncatedNormal.. autoclass:: UQpy.distributions.collection.TruncatedNormal
Uniform distribution having probability density function
f(x|a, b) = \dfrac{1}{b-a}
where a=loc and b=loc+scale
The :class:`.Uniform` class is imported using the following command:
>>> from UQpy.distributions.collection.Uniform import Uniform.. autoclass:: UQpy.distributions.collection.Uniform