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Conditional Mean and Variance

Note in the latter part of HC 2.2 how to find the conditional mean and conditional variance. The first job is to find the conditional density. Important results are:

$\displaystyle E\{E(Y\vert X)\}$ $\textstyle =$ $\displaystyle E(Y)$ (2.4)
$\displaystyle \mbox{var}\{E(Y\vert X)\}$ $\textstyle \leq$ $\displaystyle \mbox{var}(Y)$ (2.5)
$\displaystyle \mbox{var}\{Y\}$ $\textstyle =$ $\displaystyle E\{\mbox{var}(Y\vert X)\} + \mbox{var}\{E(Y\vert X)\}$ (2.6)

The proof of 2.6 follows from basic definitions:-

\begin{eqnarray*} % latex2html id marker 788E\{\mbox{var}(Y\vert X)\} &=& E\l... ...}(Y)& =& E[\mbox{var}(Y\vert X)] + \mbox{var}[E(Y\vert X)] \ \ . \end{eqnarray*}




Bob Murison 2000-10-31