Thread: Conditional expectation of t-distribution given chi sq?

1. Conditional expectation of t-distribution given chi sq?

Given that $U$ ~ $\chi^{2}_{n-1}$, $T$ ~ $t_{n-1}$ and $Z$ ~ $N(0,1)$, I need to find $E(T|U)$ and $Var(T|U)$ and hence deduce $E(T)$ and $Var(T)$ using iterated expectation and variance formulae.

I know that $T = \frac{Z}{\sqrt{\frac{U}{n-1}}}$

Surely, I should just be able to find $E(T)$ and $Var(T)$ without going through $E(T|U)$ and $Var(T|U)$, shouldn't I? Since I know $E(Z), Var(Z), E(U)$ and $Var(U)$.

How would I work out $E(T|U)$ and $Var(T|U)$, and is there a reason why I would need to work these out before coming to my answers for $E(T)$ and $Var(T)$?

2. $\mathrm E(T|U)=\frac{\mathrm E(Z)}{\sqrt{\frac{U}{n-1}}}=0$ so that $\mathrm E(T)=\mathrm E(\mathrm E(T|U))=0$.

$\mathrm{Var}(T|U)=\frac{\mathrm{Var}(Z)}{\frac{U}{ n-1}}=\frac{n-1}U$. Hence $\mathrm{Var}(T)=\mathrm E(\mathrm{Var}(T|U))+\mathrm{Var}(\mathrm E(T|U))=(n-1)\mathrm E(1/U)=\frac{n-1}{n-3}$ (assuming $n>3$).

Pray tell, how do you propose to work these out merely using the mean and variance of $Z$ and $U$?