Random variable

A random variable is a measurable function $X:(\Omega,\mathcal{F})\to(\mathbb{R},\mathcal{B}(\mathbb{R}))$ defined on a probability space $(\Omega,\mathcal{F},\mathbb{P})$, meaning that for every Borel set $B\subseteq\mathbb{R}$ one has $X^{-1}(B)\in\mathcal{F}$.

A random variable induces a distribution (law) on the real numbers via $\mathbb{P}_X(B)=\mathbb{P}(X\in B)$, and quantities such as expectation and variance are defined from this law.

Examples:

• For an event $A\in\mathcal{F}$, the indicator $X=\mathbf{1}_A$ (equal to $1$ on $A$ and $0$ on $A^c$) is a random variable.
• In a fair coin toss space, the function $X(H)=1$ and $X(T)=0$ is a random variable taking values in $\{0,1\}$.