Consider the regionSof the plane given by

S= [1,1][2,2][1,2][1,1]

i.e.Sis the union of the rectangle with vertices (1,2),(1,2),(1,2),(1,2) and the rectangle with vertices (1,1),(1,1),(2,1),(2,1).

LetX, Ybe two random variables with a joint uniform PDFfX,Y.

(a)Find the conditional PDFsfY|X(y|x) andfX|Y(x|y) for various relevant values ofxandy.

(b) FindE(X|Y),E(X) and var(X|Y). Use these to calculate var(X).

(c) FindE(Y|X),E(Y) and var(Y|X). Use these to calculate var(Y).