Let X be a normal random variable with parameters = 0 and 2 = 1, and let I, independent of X, be such

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} = 1/2 = P{I = 0}. Now define Y by

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.

X =1-X if I =1 if I = 0