1Let X and Y be independent N0 1 random variables and define

1.Let X and Y be independent N(0, 1) random variables, and define a new random variable Z by

Z =X if XY > 0

X if XY< 0

Question: Show that Z has normal distribution.

Show that the joint distribution of Z and Y is not bivariate normal. (Hint: Note that Z and Y have the same sign. Argue why this is a sufficient condition to proveyour result.)

Solution

Note that if we choose Y = X, then the product is XY is always greater than 0 and that P(Z<0) is always 0. It in turn means that Z can\'t assume any negative value and hence it can\'t follow the normal distribution.

Though the student has asked for Z to be proven as to follow a normal distribution, it can be seen from the definition that Z can\'t possibly follow a normal distribution.