class |
BivariateNormalDistribution |
The bivariate normal distribution is a continuous probability distribution
of two variables, $x$ and $y$, with cdf
$$
\begin{align*}
M(x, y, \rho) = \frac{1}{2\pi\sqrt{1 - \rho^2}}\int_{-\infty}^x\int_{-\infty}^{y} e^{\frac{-(X^2 - 2\rho XY + Y^2)}{2(1 - \rho^2)}} dX dY
\end{align*}
$$
where $\rho$ is the correlation between $x$ and $y$.
|
class |
ChiSquareDistribution |
A $\chi^2$ distribution with $k$ degrees of freedom is the distribution of
the sum of squares of $k$ independent standard normal random variables with
cdf and inverse cdf
$$
\begin{align*}
F(x) &=\frac{\gamma\left(\frac{k}{2}, \frac{x}{2}\right)}{\Gamma\left(\frac{k}{2}\right)}\\
F^{-1}(p) &= 2\gamma^{-1}\left(\frac{k}{2}, p\right)
\end{align*}
$$
where $\gamma(y, z)$ is the lower incomplete Gamma function and $\Gamma(y)$
is the Gamma function.
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class |
GammaDistribution |
The Gamma distribution is a continuous probability distribution with cdf
$$
\begin{align*}
F(x)=\frac{\gamma\left(k, \frac{x}{\theta}\right)}{\Gamma(k)}
\end{align*}
$$
and pdf
$$
\begin{align*}
f(x)=\frac{x^{k-1}e^{-\frac{x}{\theta}}}{\Gamma{k}\theta^k}
\end{align*}
$$
where $k$ is the shape parameter and $\theta$ is the scale parameter.
|
class |
GeneralizedExtremeValueDistribution |
The generalized extreme value distribution is a family of continuous probability distributions that combines the Gumbel (type I),
Fréchet (type II) and Weibull (type III) families of distributions.
|
class |
GeneralizedParetoDistribution |
Calculates the Pareto distribution.
|
class |
LaplaceDistribution |
The Laplace distribution is a continuous probability distribution with probability density function
$$
\begin{align*}
f(x)=\frac{1}{2b}e^{-\frac{|x-\mu|}{b}}
\end{align*}
$$
where $\mu$ is the location parameter and $b$ is the scale parameter.
|
class |
NonCentralChiSquaredDistribution |
The non-central chi-squared distribution is a continuous probability
distribution with probability density function
$$
\begin{align*}
f_r(x) = \frac{e^-\frac{x + \lambda}{2}x^{\frac{r}{2} - 1}}{2^{\frac{r}{2}}}\sum_{k=0}^\infty
\frac{(\lambda k)^k}{2^{2k}k!\Gamma(k + \frac{r}{2})}
\end{align*}
$$
where $r$ is the number of degrees of freedom, $\lambda$ is the
non-centrality parameter and $\Gamma$ is the Gamma function ( GammaFunction). |
class |
NormalDistribution |
The normal distribution is a continuous probability distribution with probability density function
$$
\begin{align*}
f(x) = \frac{1}{\sqrt{2\pi}\sigma} e^{-\frac{(x - \mu)^2}{2\sigma^2}}
\end{align*}
$$
where $\mu$ is the mean and $\sigma$ the standard deviation of
the distribution.
|
class |
StudentTDistribution |
Student's T-distribution is a continuous probability distribution with probability density function
$$
\begin{align*}
f(x) = \frac{\Gamma\left(\frac{\nu + 1}{2}\right)}{\sqrt{\nu\pi}\Gamma(\left(\frac{\nu}{2}\right)}\
left(1 + \frac{x^2}{\nu}\right)^{-\frac{1}{2}(\nu + 1)}
\end{align*}
$$
where $\nu$ is the number of degrees of freedom and $\Gamma$ is the Gamma function ( GammaFunction). |