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Math Help - Conditional expectation of t-distribution given chi sq?

  1. #1
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    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)?
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  2. #2
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    \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?
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