(Marginal normality does not imply bivariate normality.) Let X and Y be indepen- dent n(0, 1) random variables, and define a new random variable Z by Z = X -X if XY > 0 if XY < 0. (a) Show that Z has a normal distribution. (b) Show that the joint distribution of Z and Y is not bivariate normal. (Hint: Show that Z and Y always have the same sign.)

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