2.4 Conditional Moments

\(\underline {X}_1/\underline {X}_2\) the moments of \(\underline {X}_1/\underline {X}_2\) are called conditional moments. In particular, \(E\big (\underline {X}_1/\underline {X}_2\big )\) and \(var(\underline {X}_1/\underline {X}_2)\) are the conditional mean vector and the conditional matrix of covariance \[E(\underline {X}_1/\underline {X}_2) = \int X_i f(\underline {X}_1/\underline {X}_2)d\underline {X}_1\] where \(X_i\) is a random valuable forming a component of \(\underline {X}_1\).

In general if \(g(\underline {X}_1\) is a function of the random vector \(\underline {X}_1\) and \(f(\underline {X}_1/\underline {X}_2)\) is the conditional pdf of \(\underline {X}_1\) given \(\underline {X}_2= \underline {X}_1^0\) then \[E\big (g(\underline {X}_1)\big ) = \int _{\underline {X}_1} g(\underline {X}_1) f(\underline {X}_1/\underline {X}_2) d\underline {X}_1\]

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