If X and Y are random variables, is
$$\mathbb E[Y|X]$$
a random variable also? Intuitively it seems to be the case, but I can't explain why.
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4Yes. Definition. – Did Sep 22 '16 at 17:27
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Huh. What if $X$ is the random variable that is identically equal to 0? Then you don't even have something well defined, do yo u? – Alan Sep 22 '16 at 17:28
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@Alan If $X$ is any constant random variable, then the sigma field generated by $X$ is the trivial sigma field. In this case, $E[Y|X]=E[Y]$, which we can still think of as a random variable, albeit constant. – kccu Sep 22 '16 at 17:30
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1The notation $\mathbb{E}[Y|X]$ is consistently confusing for people. It's a worthy question and should not be closed. – Caleb Stanford Sep 22 '16 at 17:31
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@kccu Thanks. Stats is one of my weakest areas, though I did take a graduate level mathematical probability class 3 years ago – Alan Sep 22 '16 at 17:33
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$f(x) = E[Y|X=x]$, the conditional expectation, is a function defined on the range of $X$. Then $f(X)$ is a random variable, called $E[Y|X]$. – Matthew Towers Sep 22 '16 at 17:48
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$E[Y | X]$ is just shorthand for $E[Y|\mathcal{F}]$ where $\mathcal{F}=\sigma(X)$ is the sigma field generated by the random variable $X$. Now $E[Y|\mathcal{F}]$ is a random variable by definition. (It is a random variable $Z$ measureable with respect to $\mathcal{F}$ such that $\int_A Z\ dP = \int_A Y \ dP$ for all $A \in \mathcal{F}$).
