I have a problem with understanding what is the difference between:
$\psi (y) := \int_{\mathbb{R}^m} x \cdot f_{X|Y}(x | y) \ dx$ ,therefore $\psi (Y)$ = $E(X \mid Y)$
and
$\phi (y) := \int_{\mathbb{R}^m} h(x) \cdot f_{X|Y}(x | y) \ dx $ ,therefore $ \phi (Y)$ = $E(h(X) \mid Y)$
Also what is the difference between $\phi (y)$ and $\phi (Y)$, is it ever equal?
Thank you very much, I would really appriciate it.
The difference between $\psi (Y)$ = $E(X \mid Y)$ and $ \phi (Y)$ = $E(h(X) \mid Y)$ is that in the first case you're considering a random variable $X$ and computing the expected value of it given the condition that you know the outcome of the random variable $Y$. Instead, in the second expression you're considering a function $h$ of the random variable, for example it can be $h(X) = X^2$ which gives you the second moment of the random variable again conditioned on $Y$. As for the second question, the two things coincide. Specifically, the integral is exactly the definition of the expected value where you have used the conditional density of the random variable.