TABLE III.

Probability distributions used and their densities. Here, the corresponding random variables are denoted by x. We use ”;” to separate random variables from parameters. For example, Normal(x;μ,σ²) means that x is the random variable (e.g. ${\int }_{-\infty }^{+\infty }dx$ Normal(x;μ,σ²) = 1), and μ and σ² are parameters characterizing this density.

Distribution	Notation	Probability density function	Mean	Variance
Normal	Normal(μ,σ2)	$\frac{1}{\sqrt{2\pi {\sigma }^2}}{e}^{-\frac{{(x-\mu )}^2}{2{\sigma }^2}}$	μ	σ 2
Symmetric Normal	SymNormal(μ,σ2)	$\frac{1}{2}\frac{e^{-\frac{{(x+\mu )}^2}{2{\sigma }^2}}}{\sqrt{2\pi {\sigma }^2}}+\frac{1}{2}\frac{e^{-\frac{{(x-\mu )}^2}{2{\sigma }^2}}}{\sqrt{2\pi {\sigma }^2}}$	0	μ2 + σ2
Exponential	Exponential(μ)	$\mu e^{-\mu x}$	$\frac{1}{\mu }$	$\frac{1}{{\mu }^2}$
Chi-square	χ2(α,2)	$\frac{1}{\text{Γ}\left(\frac{\alpha }{2}\right)2^{\frac{\alpha }{2}}}{x}^{\frac{\alpha }{2}-1}{e}^{-\frac{x}{2}}$	α	2α
Gamma	Gamma(α,β)	$\frac{1}{\text{Γ}(\alpha ){\beta }^{\alpha }}{x}^{\alpha -1}{e}^{-\frac{x}{\beta }}$	$\alpha \beta$	$\alpha {\beta }^2$
Inverse-Gamma	InvGamma(α,β)	$\frac{{\beta }^{\alpha }}{\text{Γ}(\alpha )}{x}^{-\alpha -1}{e}^{-\frac{\beta }{x}}$	$\frac{\beta }{\alpha -1}$	$\frac{{\beta }^2}{{(\alpha -1)}^2(\alpha -2)}$
Beta	Beta(α,β)	$\frac{\text{Γ}(\alpha +\beta )}{\text{Γ}(\alpha )\text{Γ}(\beta )}{x}^{\alpha -1}{(1-x)}^{\beta -1}$	$\frac{\alpha }{\alpha +\beta }$	$\frac{\alpha \beta }{{(\alpha +\beta )}^2(\alpha +\beta +1)}$
Bernoulli	Bernoulli(q)	$(q-1){\delta }_0(x)+q{\delta }_1(x)$	$q$	$q(1-q)$