Question: Suppose that \(Z\) has a standard normal distribution and that \(Y\) is an independent \(\chi^{2}\) -distributed random variable with \(v\) df. Then, according to Definition 7.2

has a \(t\) distribution with \(v\) df.

a If \(Z\) has a standard normal distribution, give \(E(Z)\) and \(E\left(Z^{2}\right)\). [Hint: For any random variable, \(\left.E\left(Z^{2}\right)=V(Z)+(E(Z))^{2} .\right]\)

\(\mathbf{b}\) if \(Y\) has a \(\chi^{2}\) distribution with \(v \mathrm{df}\),

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

i \(\quad E(T)=0\), if \(v>1\)

ii \(\quad V(T)=v /(v-2)\), if \(v>2\)

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