Chapter 5, Problem 156SE

Refer to Exercise 5.86. Suppose that Z is a standard normal random variable and that Y is an independent χ² random variable with ν degrees of freedom.

1. a Define $W=Z/\sqrt{Y}.$ Find Cov(Z, W). What assumption do you need about the value of ν?
2. b With Z, Y, and W as above, find Cov(Y, W).
3. c One of the covariances from parts (a) and (b) is positive, and the other is zero. Explain why.

5.86 Suppose that Z is a standard normal random variable and that Y₁ and Y₂ are χ²-distributed random variables with ν₁ and ν₂ degrees of freedom, respectively. Further, assume that Z, Y₁, and Y₂ are independent.

1. a Define $W=Z/\sqrt{Y_1}.$ Find E(W) and V(W). What assumptions do you need about the value of ν₁? [Hint: $W=Z\left(1/\sqrt{Y_1}\right)$ = g(Z)h(Y₁). Use Theorem 5.9. The results of Exercise 4.112(d) will also be useful.]
2. b Define U = Y₁/Y₂. Find E(U) and V(U). What assumptions about ν₁ and ν₂ do you need? Use the hint from part (a).