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In working out a proof, I come to $var(Y + W \space|\space W)$ where $W$ and $Y$ are both random variables. Since $W$ is given, does that mean that the following is true? I would appreciate an explanation as to whether or not the following is true. Thank you in advance.

$var(Y + W \space|\space W) = var(Y)$

user3131893
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1 Answers1

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You know that $E[Y +W \mid W] = E[Y \mid W] + W$ because the expected value is linear. Therefore,

$$Var(Y+W \mid W) = E\Big [\Big((Y+W)-E[Y+W \mid W]\Big)^2 \mid W\Big] \\ = E[(Y-E[Y\mid W])^2\mid W] = Var(Y \mid W)$$

benh
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