1) Let X, Y,Z have joint pdf f(x, y, z) = 2(x + y + z)/3, 0 < x < 1, 0 < y <1, 0 < z < 1, zero elsewhere. (a)* Compute P(1/3 < X < 1/2 , 1/3 < Y < 1/2 , 1/3 < Z < 1/2). (b)* Calculate E(XYZ² + X²Y²) (c)* pdf of Y only. (d)* Find the conditional distribution of X and Y, given Z = z, and evaluate E(XY|Z=z).

2) Let X and Y be independent random variables with means ₁, ₂, and variances ²₁, ²₂. Determine the correlation coefficient of X and Z = XY in terms of ₁, ₂, ²₁, ²₂.