Only 4.9.

4.8. (Example of a nonnormal bivariate distribution with normal marginals.) Let X, be . N(0, 1), and let" X2 = -X, if-IsX=1 X, otherwise Show each of the following. (a) X2 also has an N(0, 1) distribution. (b) X1 and X2 do not have a bivariate normal distribution. Hint: (a) Since X1 is N(0, 1), P[-1 1] = .3174.4.9. Refer to Exercise 4.8, but modify the construction by replacing the break point 1 by . c so that X2= -X1 if-c=Xisc X, elsewhere Show that c can be chosen so that Cov (X1, X2) = 0, but that the two random variables are not independent. Hint: For c = 0, evaluate Cov ( X1 , X2) = E[ X,(X,)] For c very large, evaluate Cov ( X], X2) = E[X,(-X]) ]