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Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $X: \Omega \rightarrow \mathcal{X}$ and $Y: \Omega \rightarrow \mathcal{Y}$ be random variables.

I have a question comparing the conditional expectations $\mathbb{E}[Y|X]$ and $\mathbb{E}[Y|X=x]$.

1.) Does $\mathbb{E}[Y|X=x]$ mean that $X$ is fixed to the value $x$? I.e., that it only varies with $Y$?

2.) From which space to which spaces do the two conditional expectations map, respectively?

3.) If $X$ and $Y$ are absolutely continuous real valued random variables and have a joint density $f_{X, Y}$ with respect to the (product) Lebesgue-measure $\lambda \otimes \lambda =\lambda^2$, that is, $\mathbb{P}_{X, Y}(B, C)=\int_{B \times C}f_{X, Y}(x, y)\lambda^2(dx, dy)=\int_B \int_C f_{X, Y}(x, y)\lambda(dx) \lambda(dy) $ with Borel sets $B$ and $C$. Then:

$$\mathbb{E}[Y|X=x] = \int y \, f_{Y|X=x} \lambda(dy) = \int y \, \frac{f_{X, Y}(x,y)}{f_X(x)}\lambda(dy)$$.

My question is now, how would that equation look like for $\mathbb{E}[Y|X]$? Does it make sense to write something like $f_{Y|X=X}(y)$ and $f_{X,Y}(X,y)$ and do these quantity even exist? I.e., should I write $$\mathbb{E}[Y|X] = \int y \, f_{Y|X=X} \lambda(dy) = \int y \, \frac{f_{X, Y}(X,y)}{f_X(X)}\lambda(dy)$$?

4.) Following the commments under the answer: Under which circumstances do the above formulations of the conditional expectation hold? And how do I compute the conditional expectation if these conditions do not hold, i.e., in the general case?

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    $\begingroup$ (1) By definition, the conditional expectation is a function of the value of $X$--not of $Y.$ (2) You can't define the conditional expectation as you do for continuous variables because $f_X(x)$ is never uniquely defined. You need a more sophisticated approach, that of projection to a sub sigma algebra. I believe stats.stackexchange.com/questions/230545 might answer all your questions. $\endgroup$
    – whuber ♦
    Oct 21, 2021 at 19:38
  • $\begingroup$ Thanks for the link, it is indeed insightful but I dont think it helps me solve all my understanding problems. In the answer, he is talking about an event A. Is X=x also an event? I mean A is a set and X=x is only a number right? $\endgroup$
    – guest1
    Oct 21, 2021 at 20:07
  • $\begingroup$ en.wikipedia.org/wiki/Conditional_expectation in the section "continuous random variables" they define the conditional expectation exactly like I do $\endgroup$
    – guest1
    Oct 21, 2021 at 20:09
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    $\begingroup$ That's an elementary approach intended to avoid the complexities of the measure-theoretic definition. But the language you use suggests you are already beyond such simplifications. $\endgroup$
    – whuber ♦
    Oct 21, 2021 at 21:28
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    $\begingroup$ It is not wrong--it is just not universally applicable. $\endgroup$
    – whuber ♦
    Oct 22, 2021 at 12:34

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