Let (Ω, Pr) be a probability space, and let X and Y be two independent random variables that are positive and have non-zero variance. (a)  Prove that X^2 and Y are independent. Note that by symmetry, it will also follow that Y^2 and X are independent. (b)  Use the result from part (a) to show that the random variables W = X + Y and Z = XY are positively correlated (i.e. Cov(W, Z) > 0).

Suppose we have three independent random variables X, Y and Z where... Var(X + 2Y) = 13, Var(2Y + 4Z) = 40, and Var(Z) = 2. a) If E(Y) = E(Z) = 1, what is E(Y2+Z2)? b) What is SD(X + Y + Z)?

3. Let X1, X2, ..., Xn be a random sample from the population N(u, o²) and Y1, Y2,...,Ym a random sample from the population N(uy, o²), where both means (ua and uy) are assumed known. Note that the two distributions have a common variance of o?. (a) If the X's and Y's are independent, show that the MLE for the common variance is: E(Xi - Ha)² +E(Y; - Hy)² n + m (b) Is the MLE found above unbiased for o??

Let Z₁ and Z₂ be independent standard normal random variables. Let pe [-1, 1]. Find a matrix L such that has a distribution. X = LZ N (1). (1))