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Normal

Normal Normal

$\frac{1}{\sigma\sqrt{2\pi}}\,e^{ -\frac{(x-\mu)^2}{2\sigma^2} }$

\[ (2\pi)^{-\frac{k}{2}}|\boldsymbol\Sigma|^{-\frac{1}{2}}\, e^{ -\frac{1}{2}(\mathbf{x}-\boldsymbol\mu)'\boldsymbol\Sigma^{-1}(\mathbf{x}-\boldsymbol\mu) } \]

Skew-Normal

\[ \frac{1}{\omega\pi} e^{-\frac{(x-\xi)^2}{2\omega^2}} \int_{-\infty}^{\alpha\left(\frac{x-\xi}{\omega}\right)} e^{-\frac{t^2}{2}}\ dt \]

Heavy-Tailed

Cauchy

\[ \frac{1}{\pi\gamma\,\left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}\ \]

Log-normal

\[\frac{1}{x\sqrt{2\pi\sigma^2}}\, e^{-\frac{\left(\ln x-\mu\right)^2}{2\sigma^2}}\]

Pareto

\[ \frac{\alpha\,x_\mathrm{m}^\alpha}{x^{\alpha+1}}\text{ for }x\ge x_m \]

Exponential

\[ \lambda e^{-\lambda x} \]

Gamma

\[ \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha \,-\, 1} e^{- \beta x } \]

Beta

\[ \frac{x^{\alpha-1}(1-x)^{\beta-1}} {\mathrm{B}(\alpha,\beta)} \]

Dirichlet

\[ \frac{1}{\mathrm{B}(\boldsymbol\alpha)} \prod_{i=1}^K x_i^{\alpha_i - 1},\boldsymbol\alpha=(\alpha_1,\ldots,\alpha_K), \mathrm{B}(\boldsymbol\alpha) = \frac{\prod_{i=1}^K \Gamma(\alpha_i)}{\Gamma\bigl(\sum_{i=1}^K \alpha_i\bigr)} \]

Poisson

\[ \frac{\lambda^k}{k!}\cdot e^{-\lambda} \]