 Conditional expectation

For an integrable random variable $X$ and a sub-sigma-algebra $\mathcal{G}$, the conditional expectation $\mathbb{E}[X\mid \mathcal{G}]$ is the almost-everywhere unique $\mathcal{G}$-measurable integrable random variable satisfyingfor every $G\in \mathcal{G}$.

Conditional expectation is a fundamental concept in probability theory and statistics that refers to the expected value of a random variable given that certain conditions or information are known. It captures the idea of updating our expectations based on additional information. Formally, if \( X \) is a random variable and \( Y \) is another random variable (or an event), the conditional expectation of \( X \) given \( Y \) is denoted as \( \mathbb{E}[X | Y] \).