Random variables - density functions

(a) If Z1 and Z2 are independent and have N(0,1) densities, let

X1 = a11Z1 + a12Z2 + c1 and let X2 = a21Z1 + a22Z2 + c2

where a11,a12,a21,a22,c1 and c2 are constants. Determine the joint density of X1 and X2.

(b) Suppose that the random variable X on (0,1) has density

f(x) = 3x2 0 < x < 1

Determine the density functions of

(i) U = X1/2

(ii) V = -log(X)

(c) Now suppose Y is uniformly distributed on (0,1) and independent of X. Determine the density of

W = max(X,Y)