# Covariance involving Expectation

1. Nov 1, 2013

### Karnage1993

1. The problem statement, all variables and given/known data
Suppose $X,Y$ are random variables and $\varepsilon = Y - E(Y|X)$. Show that $Cov(\varepsilon , E(Y|X)) = 0$.

2. Relevant equations
$E(\varepsilon) = E(\varepsilon | X) = 0$
$E(Y^2) < \infty$

3. The attempt at a solution
$Cov(\varepsilon , E(Y|X)) = E((\varepsilon - E(\varepsilon)(E(Y|X) - E(E(Y|X)))) = E(\varepsilon (E(Y|X) - E(Y))) = E(\varepsilon E(Y|X)) - E(\varepsilon E(Y))$

From there, I didn't know what to do next so I tried it again using the formula for covariance:

$Cov(\varepsilon , E(Y|X)) = E(\varepsilon E(Y|X)) - E(\varepsilon )E(E(Y|X)) = E(\varepsilon E(Y|X))$

Which means $E(\varepsilon E(Y|X)) - E(\varepsilon E(Y) = E(\varepsilon E(Y|X))$ or in other words, $E(\varepsilon E(Y)) = 0$. I'm not sure what to do now.

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