2.13 X is a Bernoulli random variable with Pr(X = 1) = 0.90, Y is distributed N(0, 4), W is distributed N(0, 16), and X, Y, and W are independent. Let S = XY + (1 - X)W. (That is, S = Y when X = 1, and S = W when X = 0.)

a. Show that E(Y 2) = 4 and E(W 2) = 16.

b. Show that E(Y3) = 0 and E(W3) = 0. (Hint: What is the skewness for a symmetric distribution?)

c. Show that E(Y4) = 3 * 42 and E(W4) = 3 * 162. (Hint: Use the fact that the kurtosis is 3 for a normal distribution.)

d. Derive E(S), E(S2), E(S3) and E(S4). (Hint: Use the law of iterated expectations conditioning on X = 0 and X = 1.)

e. Derive the skewness and kurtosis for S.