Random Variables

Expectation

For a discrete random variable, $\mathbb{E}[X] = \sum_x x \, P(X = x)$.

Variance

$$\operatorname{Var}(X) = \mathbb{E}\big[(X - \mathbb{E}[X])^2\big] = \mathbb{E}[X^2] - \mathbb{E}[X]^2.$$

Common distributions

Distribution	Mean	Variance
Bernoulli$(p)$	$p$	$p(1-p)$
Binomial$(n, p)$	$np$	$np(1-p)$
Normal$(\mu, \sigma^2)$	$\mu$	$\sigma^2$