1. Let Y1 and Y2 be independent random variables that are both uniformly distributed on the interval (0,1). Find P(Y1< 2*Y2|Y1< 3*Y2).

2. Let Z ~ N(0,1) and Y ~ χ2(ν) (Chi-square with ν degrees of freedom). Assume that Z is independent of Y and let W = Z/√Y . Obtain E(W) and Var(W).

3. The joint density of Y1 and Y2 is given by

f(y1,y2) = {y1+ y2, 0≤ y1≤ 1, 0 ≤ y2≤ 10

O therwise.

Obtain Var(30*Y1+ 25*Y2).

Thank you very much.