Probability Distributions

\text{Gaussian:}\quad f(x) = \frac{1}{\sigma\sqrt{2\pi}}\exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)

\text{Binomial:}\quad P(X=k) = \binom{n}{k} p^{k}(1-p)^{n-k}

\text{Poisson:}\quad P(X=k) = \frac{\lambda^{k} e^{-\lambda}}{k!}

\text{Exponential:}\quad f(x) = \lambda e^{-\lambda x},\quad x \ge 0

\text{Beta:}\quad f(x;\alpha,\beta) = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}

\text{Multivariate Gaussian:}\quad f(\mathbf x) = \frac{1}{(2\pi)^{d/2}|\Sigma|^{1/2}}\exp\!\left(-\tfrac{1}{2}(\mathbf x - \boldsymbol\mu)^\top \Sigma^{-1}(\mathbf x - \boldsymbol\mu)\right)