How to Solve Multivariate Probability — Multivariate — conditional expectation / variance Questions on Exam P
Sample Practice Problem
ID: #6WF04
Let \(X\) be exponential with rate \(2\). Independently, let \(Z\) be exponential with rate \(4\), and define \(Y=X+Z\). Calculate \(\operatorname{Var}(\operatorname{E}[Y\mid X])\).
📖 Worked Solution & Strategy
The conditional representation separates \(Y\) into the observed value of \(X\) and an independent exponential gap. Conditional expectation and variance therefore isolate exactly which source of randomness remains after \(X\) is known. Because \(Z\) is independent of \(X\), conditional expectation gives \(\operatorname{E}[Y\mid X]=X+\operatorname{E}[Z\mid X]=X+\operatorname{E}[Z]=X+1/4\). Adding the constant \(1/4\) does not alter variance, so \(\operatorname{Var}(\operatorname{E}[Y\mid X])=\operatorname{Var}(X)=1/2^2=\frac{1}{4}\).