Distribution (law) - Knowlpedia

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

\mu_X(A)=\mathbb P(X\in A)\qquad\text{for all }A\in\mathcal S,

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

Remarks

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 $[0,1]$ ).

Representations

On a countable state space a law is described by a probability mass function. A law on the real line is always determined by its cumulative distribution function; when it has a probability density function, probabilities can also be calculated by integration.