X is a Bernoulli random variable with Pr(X = 1) = 0.90, Y is
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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, a
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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.
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