I'm trying to confirm an answer I saw in another post here and prove that $ E[Z|X>\gamma]=∫_\gamma^\infty E[Z∣X=x]f_X(x)\text{d}x $ where $X$ and $Z$ are random variables with density functions $f_X$ and $f_Z$ respectively and $\gamma$ is a constant.
Unfortunately I keep getting an extra $\text{P}(X>\gamma)^{-1}$ factor on the right hand side: $ E[Z|X>\gamma]=\text{P}(X>\gamma)^{-1}∫_\gamma^\infty E[Z∣X=x]f_X(x)\text{d}x $
This extra factor comes from calculating $f_{Z|X>\gamma}$ = $\text{P}(X>\gamma)^{-1}\int_{\gamma}^{\infty} f_{X,Z}(x,z) \text{d}x$ . Is this correct or is the original answer wrong?
You are correct and the other answer is correct, you are just not acquainted with the notation. $E(Z; A) $ is a notation for $E(Z1_A)$ which is equal to $E(Z|A)P(A).$