Question

2. (10 pts) Random variables X and Y have the following joint PDF: 0.1, if both 11 and 2S2 Jx( if both Is2 ad Sys; 0, otherwise. (a) Prepare neat, fully labeled sketches of xir (r) (b) Find EKİY=y] and var(X|Y-v). (c) Find E[x (d) Find var(x)using the law of conditional variances.

Answer 1

$(b)~E(X|-1\leq Y\leq 1)=\int_{-1}^{2}x/3dx=\frac{1}{6}(2^2-(-1)^2)=\frac{1}{2}\\ E(X|-2\leq Y<-1~or~1<Y\leq 2)\int_{-1}^{1}x/2dx=0\\ E(X^2|-1\leq Y\leq 1)=\int_{-1}^{2}x^2/3dx=\frac{1}{9}(2^3-(-1)^3)=1\\ E(X^2|-2\leq Y<-1~or~1<Y\leq 2)\int_{-1}^{1}x^2/2dx=\frac{1}{4}(2)=\frac{1}{2}\\ Var(X|-1\leq Y\leq 1)=1-(1/2)^2=3/4\\ Var(X|-2\leq Y<-1~or~1<Y\leq 2)=1/2\\ (d)~Var(X)=E(Var(X|Y))+Var(E(X|Y))\\ =0.3*\frac{3}{4}\int_{-1}^1dy+0.2*\frac{1}{2}\left (\int_{-2}^{-1}dy+\int_{1}^2dy \right )\\ =0.9/2+0.2=0.65,~here,~Var(E(X|Y))=0\\ (c)~E(X)=E(E(X|Y))=\frac{0.3}{2}\int_{-1}^1dy=0.3$