Distribution

Definition

${\mu }_X:=P\circ X^{-1}:\mathcal{R}\to [0,1]$ where $\mathcal{R}$ is a Borel Sigma-Field

A distribution ${\mu }_X$ is a function for the Random Variable $X$ on Probability Space $(\Omega ,\mathcal{F},P)$

Facts

${\mu }_X$ is a Probability Measure on $(\mathbb{R},\mathcal{R})$, so $(\mathbb{R},\mathcal{R},{\mu }_X)$ is a Probability Space. The Probability Space $(\mathbb{R},\mathcal{R},{\mu }_X)$ is called the Probability Space induced by $X$