Probability distributions

Bernoulli: \(X\in\{0,1\}\), \(P(X=1)=p\), mean \(p\), variance \(p(1-p)\).

Binomial: \(X\sim\operatorname{Bin}(n,p)\), mean \(np\), variance \(np(1-p)\).

Poisson: \(X\sim\operatorname{Pois}(\lambda)\), \(P(X=k)=e^{-\lambda}\lambda^k/k!\), mean and variance \(\lambda\).

Exponential: \(T\sim\operatorname{Exp}(\lambda)\), density \(f(t)=\lambda e^{-\lambda t}\) for \(t\ge0\), mean \(1/\lambda\).

Normal: \(X\sim N(\mu,\sigma^2)\), with mean \(\mu\) and variance \(\sigma^2\).

Uniform: \(U\sim U(0,1)\), commonly used to generate stochastic events computationally.