Overview

This topic is Bertsekas's — the toolkit that turns the single-random-variable machinery of the previous two topics into something that can actually handle a portfolio, a basket, or an aggregated risk. It answers five distinct questions: how do you find the distribution of a transformed random variable ($Y=g(X)$, via the CDF method and, for monotonic $g$, a direct formula)? How do two random variables' fluctuations co-move (covariance and correlation)? What does it mean to treat $\mathbb{E}[X\mid Y]$ as a random variable in its own right, and what does that buy you (the law of iterated expectations, the law of total variance)? How does the moment generating function unify moment calculation and distributional identification into one tool? And what happens when the number of terms in a sum is itself random? The topic closes with Jensen's inequality — the single convexity fact used throughout the rest of the vault to reason about option payoffs and risk-averse utility.