This is a re-question with maybe another view on finding probability density function of a conditional expectation for same "a)".
That is, which one is the density function of a conditional expectation?
The answers there have stated that density of $E(X|Y)$ is just $f_Y(y)$. Is that a general true?
I mean, as stated $$ \begin{align} Z=E(X|Y) & \Rightarrow P(Z(Y) = z) = P(Y = z^{-1}(z)) = P(Y = y(z)) \\ & \Rightarrow f_Z(z) = f_Y(y(z)) \end{align} $$ for the particular case, the density of $Z$ should be $f_Y(y)$, always?
Related question: Is $E(X|Y)$ a bijection from $Y$ to $Z$?
Thanks.
No. $\mathsf E(X\mid Y)$ is $Z$. However, as such, $Z$ is measurable on the $\sigma$-algebra for $Y$.
It is not generally the case that there will be a bijection between the support for the probability mass function of $Y$ and that of $Z$.
So when $\mathsf E(X\mid Y)= g(Y)$ for some function $g$ which happens to be a bijection on the support for the probability mass function of $Y$, then: $$\mathsf P_{g(Y)}(z)=\mathsf P_Y(g^{-1}(z))$$
This does not work when $g$ is not invertable.