Let $X$ be a \reftext{def:probability-space-random-variable-2026a}{random variable} on a probability space $(\Omega,\mathcal{F},P)$.

The \textbf{distribution} (or \textbf{law}) of $X$ is the function

on the \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel $\sigma$-algebra}. It is a probability \reftext{def:measure-measure-space-2026a}{measure} on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$: $P_X(\varnothing)=P(\varnothing)=0$, $P_X(\mathbb{R})=P(\Omega)=1$, and countable additivity holds because preimages of pairwise disjoint sets are pairwise disjoint and taking preimages commutes with countable unions.

The \textbf{cumulative distribution function} of $X$ is

Two random variables (possibly on different probability spaces) are \textbf{identically distributed} if their distributions are equal.