Distribution (law)

A distribution (law) of a random variable $X:\Omega\to S$ (more generally, a measurable map into a measurable space $(S,\mathcal S)$) is the probability measure $\mu_X$ on $(S,\mathcal S)$ defined by

where $X\in A$ abbreviates the event $\{\omega\in\Omega: X(\omega)\in A\}\in\mathcal F$.

This is the pushforward of $\mathbb P$ along $X$; it packages all probabilities of measurable sets in the state space into a single measure, and is often written $\mu_X=\mathbb P\circ X^{-1}$.

Examples:

• If $X$ is Bernoulli$(p)$ taking values in $\{0,1\}$, then $\mu_X(\{1\})=p$ and $\mu_X(\{0\})=1-p$.
• If $X$ is uniform on $[0,1]$, then $\mu_X((a,b))=b-a$ for $0\le a<b\le 1$ (equivalently, $\mu_X$ has density $1$ with respect to Lebesgue measure on $[0,1]$).