Equal in Distribution

Last updated Nov 1, 2022

Let $X, Y$ be Random Variables and let $F_{X}, F_{Y}: \mathbb{R} \to \mathbb{R}$ be their respective Probability Distribution Functions. Then $X \overset{\text{d}}{=} Y$ ($X, Y$ are Equal in Distribution), if $F_{X}$ = $F_{Y}$.

1. If two Random Variables are Equal in Distribution, then their induced Induced Probability Measures on $\mathbb{R}$ must be the same because of the Correspondence of Lebesgue-Stieltjes Measures and Distribution Functions.

2. Suppose $X \overset{d}{=} Y$. Overload $F_{X} = F = F_{Y}$ as both the Distribution Functions and induced Induced Probability Measures for $X, Y$ respectively. Then for some Borel Measureable Function $h: \mathbb{R} \to \mathbb{R}$, we have that

   $$\int\limits_{A} h(x)F_{X}(dx) = \int\limits_{A} h(x)F(dx) = \int\limits_{A} h(y)F_{Y}(dy)$$ In particular $\forall k \in \mathbb{N}$ $$\mathbb{E}|X|^{k} = \mathbb{E}|Y|^{k}$$ That is, $X$ has the same Moments as $Y$.