Introduction to Econometrics 3rd Edition by James Stock, Mark Watson

تمرين 27

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 ( Y 2 ) = 1 and E ( W 2 ) = 100.
b. Show that E ( Y 3 ) = 0 and E ( W 3 ) = 0.
c. Show that E ( Y 4 ) = 3 and E ( W 4 ) = 3 × 100 2.
d. Derive E ( S ) , E ( S 2 ) , E ( S 3 ) and E ( S 4 ).
e. Derive the skewness and kurtosis for S.