How to Solve Multivariate Probability — Multivariate — joint density, probability over a region Questions on Exam P
Sample Practice Problem
ID: #8ZH5S
\(X\) is exponential with rate \(\frac{1}{2}\). Given \(X=x\), \(Y\sim\operatorname{Uniform}(x,x+1)\). Calculate \(\Pr(X+Y<2)\).
📖 Worked Solution & Strategy
Write \(Y=X+U\), where \(U\sim\operatorname{Uniform}(0,1)\). Now \(X+Y=2X+U\). All \(U\in(0,1)\) work for \(0<X<1/2\); for \(1/2<X<1\), the allowed fraction is \(2-2X\). Hence the probability is \(\int_0^{1/2}\frac{1}{2}e^{-\frac{1}{2}x}\,dx+\int_{1/2}^1(2-2x)\frac{1}{2}e^{-\frac{1}{2}x}\,dx\). Therefore, \(\Pr(X+Y<2)\;=0.3109\), which matches the correct option.