Kolmogorov axioms

A probability space $(\Omega, \mathcal{F}, P)$ consists of a sample space $\Omega$, a $\sigma$-algebra $\mathcal{F}$, and a measure $P$ satisfying:

1. $P(A) \geq 0$ for all $A \in \mathcal{F}$
2. $P(\Omega) = 1$
3. $P(\bigcup_{i=1}^\infty A_i) = \sum_{i=1}^\infty P(A_i)$ for disjoint events

Key distributions

Distribution	PMF/PDF	Mean	Variance
Bernoulli$(p)$	$p^x(1-p)^{1-x}$	$p$	$p(1-p)$
Binomial$(n,p)$	$\binom{n}{k}p^k(1-p)^{n-k}$	$np$	$np(1-p)$
Poisson$(\lambda)$	$\lambda^k e^{-\lambda}/k!$	$\lambda$	$\lambda$
Exponential$(\lambda)$	$\lambda e^{-\lambda x}$	$1/\lambda$	$1/\lambda^2$
Gaussian$(\mu,\sigma^2)$	see gaussian-distribution	$\mu$	$\sigma^2$

Expectation and variance

\[\mathbb{E}[X] = \int x\, dP(x), \qquad \text{Var}(X) = \mathbb{E}[X^2] - \mathbb{E}[X]^2\]

The law of total expectation: $\mathbb{E}[X] = \mathbb{E}[\mathbb{E}[X \mid Y]]$.