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

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

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

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

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

    How would I work out $\displaystyle E(T|U)$ and $\displaystyle Var(T|U)$, and is there a reason why I would need to work these out before coming to my answers for $\displaystyle E(T)$ and $\displaystyle Var(T)$?
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  2. #2
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    $\displaystyle \mathrm E(T|U)=\frac{\mathrm E(Z)}{\sqrt{\frac{U}{n-1}}}=0$ so that $\displaystyle \mathrm E(T)=\mathrm E(\mathrm E(T|U))=0$.

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

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