In Exercise 5.65, we considered random variables Y1 and Y2 that, for −1 ≤α ≤ 1, have joint density function given by

We established that the marginal distributions of Y1 and Y2 are both exponential with mean 1 and showed that Y1 and Y2 are independent if and only if α = 0. In Exercise 5.85, we derived E(Y1Y2).

a Derive Cov(Y1,Y2).

b Show that Cov(Y1,Y2) = 0 if and only if α = 0.

c Argue that Y1 and Y2 are independent if and only if ρ = 0.

Exercise 5.65

Suppose that, for −1 ≤ α ≤ 1, the probability density function of (Y1, Y2) is given by

a Show that the marginal distribution of Y1 is exponential with mean 1.

b What is the marginal distribution of Y2?

c Show that Y1 and Y2 are independent if and only if α = 0.

Notice that these results imply that there are infinitely many joint densities such that both marginals are exponential with mean 1.