Real-Valued Random Variable Expectation

Definition

The expectation (or expected value) of a real-valued random variable defined on a probability space is its integral with respect to the probability measure. The expectation of a random variable is also called its mean.

Notation

Suppose $f: X \to \R $ is a random variable on a probability space $(X, \mathcal{A} , P)$. We denote the expectation of $f$ by $\E f$, so that

\[ \E f := \int f dP. \]