Discrete Random Variables & Expectation

Random Variables as Measurable Maps

A discrete random variable $X$ is a measurable function $X: \Omega \to \mathbb{R}$ that maps elements from our sample space onto discrete real numbers.

Linearity of Expectation

Regardless of independence, the expected value operator satisfies strict linearity:

$\mathbb{E}[aX + bY] = a\mathbb{E}[X] + b\mathbb{E}[Y]$