X is a Bernoulli random variable with Pr(X = 1) = 0.99, Y is distributed N(0, 1),

Question:

X is a Bernoulli random variable with Pr(X = 1) = 0.99, Y is distributed N(0, 1), W is distributed N(0, 100), 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(Y2) = 1 and E(W2) = 100.
(b) Show that E(Y3) = 0 and E(W3) = 0.
(c) Show that E(Y4) = 3 and E(W4) = 3 × 1002.
(d) Derive E(S), E(S2), E(S3) and E(S4).
(e) Derive the skewness and kurtosis for S.