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I know that the law of total variance states $$Var(X)=\Bbb E[Var(X|Y)]+Var(\Bbb E[X|Y])$$ But how does one treat $Var(X|Y)$ and $\Bbb E[X|Y]$ as random variables? For example, say we know that $$\Bbb E[X|Y=y]=y \ \ \ \text {and} \ \ \ Var(X|Y=y)=1$$ I take it that directly calculating the expected value of $x$ and the variance of $1$ is not possible. So how does one actually do this practically?

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The idea is that the expectation will be some function of $y$ i.e. $\mathbb{E}[X|Y=y] = f(y)$. In these cases, expectation and variance decomposition lemmas are useful.

Take ANOVA for e.g.

$Var(X)=\Bbb E[Var(X|Y)]+Var(\Bbb E[X|Y])$ is nothing but cross-group variance - $\Bbb E[Var(X|Y)]$ + within-group variance - $Var(\Bbb E[X|Y])$.

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Don't confuse $\Bbb E[X|Y]$ and $\Bbb E[X|Y=y]$

$\Bbb E[X|Y]$ is a random variable, function of the random variable $Y$, $$\Bbb E[X|Y]=g(Y)$$ $\Bbb E[X|Y=y]$ is a number, $$\Bbb E[X|Y=y]=g(y)$$ You can compute both if you know the joint distribution of $X$ and $Y$, same with variance $\Bbb Var[X|Y]$ (random variable) or $\Bbb Var[X|Y=y]$ (number).

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  • $\begingroup$ If I have $\Bbb E[X|Y=y]$ and the distribution of $Y$, how do I establish what $\Bbb E[X|Y]$ is? Or $Var(X|Y)$ for that matter? $\endgroup$ – mathenthusiast May 22 at 11:42
  • $\begingroup$ $\newcommand{\E}{\mathbb{E}}$Once you have the expression for $\E[X\mid Y=y]$, just replace every $y$ you see in this expression with $Y$ and that is the random variable $\E[X\mid Y]$. For example, if you know that $\E[X\mid Y=y]=y$ for all $y$, then $\E[X\mid Y] = Y$. As another example, if you knew that $\E[X\mid Y=y] = y^3$ for all $y$, then $\E[X\mid Y]$ is the random variable $Y^3$. A similar story holds for $\operatorname{Var}(X\mid Y)$. $\endgroup$ – Minus One-Twelfth May 22 at 12:58

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