2.13 X is a Bernoulli random variable with Pr1X = 12 = 0.90; Y is distributed N(0, 4); W is distributed N(0, 16); and X, Y, and W are independent.

Let S = XY + 11 - X2W. (That is, S = Y when X = 1, and S = W when X = 0.)

a. Show that E1Y22 = 4 and E1W22 = 16.

b. Show that E1Y32 = 0 and E1W32 = 0. (Hint: What is the skewness for a symmetric distribution?)

c. Show that E1Y42 = 3 * 42 and E1W42 = 3 * 162. (Hint: Use the fact that the kurtosis is 3 for a normal distribution.)

d. Derive E1S2, E1S22, E1S32, and E1S42. (Hint: Use the law of iterated expectations conditioning on X = 0 and X = 1.)

e. Derive the skewness and kurtosis for S.