Probability Distributions

Discrete random variables

A discrete random variable $X$ takes specific values with probabilities that sum to $1$: $\sum P(X = x) = 1$. The expectation (mean) and variance are

$$E(X) = \sum x\,P(X = x), \qquad \operatorname{Var}(X) = \sum x^2 P(X = x) - \big(E(X)\big)^2$$