41. Let X be a normal random variable with parameters = 0 and = 1 and let I, independent of X, be such

41. Let X be a normal random variable with parameters µ = 0 and σ² = 1 and let I, independent of X, be such that P{I = 1} = P{I = 0}. Now define Y by X if I = 1 Y =

-X if I = 0 In words, Y is equally likely to equal either X or -X.

(a) Are X and Y independent?

(b) Are I and Y independent?

(c) Show that Y is normal with mean 0 and variance 1.

(d) Show that Cov(X, Y) = 0.