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(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.)

(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.)

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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(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 X if XY > 0 = { ² X if XY <0. -X Z = (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.)
Transcribed Image Text:(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 X if XY > 0 = { ² X if XY <0. -X Z = (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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