Random Variable

Definition

$X:\Omega \to \mathbb{R}\text{s.t.}\forall B\in \mathcal{R},\exists \omega \in \Omega ,\text{s.t.}{X}^{-1}(B)=\{\omega :X(\omega )\in B\}\in \mathcal{F}$

A random variable is a function $X:\Omega \to \mathbb{R}$ whose inverse function $X^{-1}(B)$ is $\mathcal{F}$-measurable for the two measurable spaces $(\Omega ,\mathcal{F})$ and $(\mathbb{R},\mathcal{R})$.

The inverse image of an arbitrary Borel set of Codomain $\mathbb{R}$ is an element of sigma field $\mathcal{F}$.

Notations

Consider a probability space $(\Omega ,\mathcal{F},\mathbb{P})$

• Outcomes: $H,T$
• Set of outcomes (Sample space): $\Omega =\{H,T\}$
• Events: $\varnothing ,\{H\},\{T\},\Omega$
• Set of events (Sigma-Field): $\mathcal{F}=\{\varnothing ,\{H\},\{T\},\{H,T\}\}$
• Probabilities: $\mathbb{P}:\mathcal{F}\to [0,1]$
• Random variable: $X:\Omega \to \mathbb{R}$

For a random variable $X$ on a Probability Space $(\Omega ,\mathcal{F},P)$

• $X\sim {\mu }_X$ if and only if the Distribution of $X$ is ${\mu }_X$
• $X\sim F_X$ if and only if the Distribution Function of $X$ is $F_X$

For a random variable $X$ on Probability Space $({\Omega }_X,{\mathcal{F}}_X,P_X)$ and another random variable $Y$ on Probability Space $({\Omega }_Y,{\mathcal{F}}_Y,P_Y)$

• $X\stackrel{d}{=}Y⟺\forall B\in \mathcal{R}:{\mu }_X(B)={\mu }_Y(B)$
• $X\stackrel{d}{=}Y⟺\forall k\in \mathbb{R}:F_X(k)=F_Y(k)$
• $X\stackrel{d}{=}Y⟺\forall k\in \mathbb{R}:P_X(X\le k)=P_Y(Y\le k)$