Given a random variable $x$, what is $Var(E[x])$?
My intuition is that it would simply be the same as $Var(x)$, but I'm not sure how to prove that. Any and all help would be greatly appreciated!
[Math] Variance of Expected Value
probabilitystatistics
Best Answer
$\mathsf E[X]$ is a constant term, so has zero variance.
To verify:
By definition: $\mathsf{Var}[N] = \mathsf E[(N-\mathsf E[N])^2]$
So then: $\mathsf {Var}[\mathsf E[X]] $ $= \mathsf E[(\mathsf E[X]- \mathsf E[\mathsf E[X]])^2] \\ = \mathsf E[0^2] \\ = 0$
Because we expect the expected value to be the expected value: $\mathsf E[\mathsf E[X]]=\mathsf E[X]$