Distribution

Last updated Nov 1, 2022

Suppose $X$ is a Random Variable on Probability Space $(\Omega, \mathcal{M}, \mathbb{P})$ with Distribution Function $F_{X}$ and Induced Probability Measure $\mu$. The Distribution is really just an overloaded shorthand for these two terms. That is, if $G$ is the Distribution then

1. $G(x) = F_{X}(x)$ $\forall x \in \mathbb{R}$
2. $G(S) = \mu(S)$ $\forall S \in \mathcal{B}(\mathbb{R})$