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.)

2. X and Y are independent random variables with X ∼ exp(λ) and Y ∼ exp(µ). It isimpossible to obtain direct observations of X and Y . Instead, we observe the randomvariables Z and W, where:Z = min {X, Y } and W =1 if Z = X, or =0 if Z = Y•

Find the joint distribution Z and W.

Prove that Z and W are independent. (Hint: Show that P(Z ≤ z|W = w) =P(Z ≤ z) for w= 0 or 1.

3.Prove the following statement that implies that the conditional expectation has optimal prediction property in terms of the mean square error.Let Y be a random variable with E(Y2) <∞. Then for any other G-measurable random variable Y˜ , E[(Y - E[Y |G])^2]≤ E[(Y - Y)^2].