Definition:Probability Distribution/General Definition

Definition

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\struct {S, \Sigma'}$ be a measurable space.

Let $X$ be a random variable on $\tuple {\Omega, \Sigma, \Pr}$ taking values in $\struct {S, \Sigma'}$.

Then the probability distribution of $X$, written $P_X$, is the pushforward $X_* \Pr$ of $\Pr$, under $X$, on $\Sigma'$.

That is:

$\ds \map {P_X} B$

$=$

$\ds \map \Pr {X^{-1} \sqbrk B}$

$\ds $

$=$

$\ds \map \Pr {\set {\omega \in \Omega : \map X \omega \in B} }$

$\ds $

$=$

$\ds \map \Pr {X \in B}$

for each $B \in \Sigma'$, where $X^{-1} \sqbrk B$ denotes the pre-image of $B$ under $X$.