Chapter 7.2, Problem 30E

Suppose that Z has a standard normal distribution and that Y is an independent χ²-distributed random variable with ν df. Then, according to Definition 7.2,

$T=\frac{Z}{\sqrt{Y/v}}$

has a t distribution with ν df.

1. a If Z has a standard normal distribution, give E(Z) and E(Z²). [Hint: For any random variable, E(Z²) = V(Z) + (E(Z))².]
2. b According to the result derived in Exercise 4.112(a), if Y has a χ² distribution with ν df, then

   $E(Y^a)=\frac{\Gamma ([v/2]+a)}{\Gamma (v/2)}{2}^a,\text{if }v>-2a.$

   Use this result, the result from part (a), and the structure of T to show the following. [Hint: Recall the independence of Z and Y.]

   1.                                 i.            E(T) = 0, if ν > 1.
   2.                               ii.            V(T) = ν/(ν − 2), if ν > 2.