Chapter 12: Conditional Expectation as a Random Variable

Learning Objectives

• Understand that the conditional expectation $\mathbb{E}[X|Y]$ is itself a random variable — a function of $Y$ — and state its key properties
• Prove that $\mathbb{E}[X|Y]$ is the minimum mean square error (MMSE) estimator of $X$ given $Y$
• State and apply the orthogonality principle: $X - \mathbb{E}[X|Y] \perp h(Y)$ for all measurable $h$
• Derive the linear MMSE estimator via the Wiener-Hopf equation and understand when LMMSE equals MMSE
• State and apply the law of total variance to decompose uncertainty into explained and unexplained components
• Connect conditional expectation to channel estimation, Wiener filtering, and Bayesian inference in wireless communications