Meaning of the expectation of a conditional variance

Question

I'm reading page 2 of conditional probability. The author states that if $\text{Var(X|Y)}$ is treated as a random variable then the expectation is $\text{E[Var(X|Y)] = E[E[X^2|Y]] - E[E(X|Y)]^2}$

My question is what exactly does the expected variance mean as far as the the random variables, X and Y are concerned? An intuitive explanation would suffice.

Answer 1

$Y$ is a random variable. $X$ is taken to depend on $Y$ in some way (could be strong, could be nonexistently weak). The variance of $X$ given $Y$ is

• for each a priori given value of $Y$,
• for the distribution of $X$ consistent with that choice of $Y$,
• what is the variance of the distribution of $X$?

Without more details about an assumed relationship between $X$ and $Y$, no more can be said that what you've quoted. (The expected variance in $X|Y$ is the difference between the expectation of (the expectation of $X^2|Y$) averaged over the distribution of $Y$s and the square of the expectation of (the expectation of $X|Y$) averaged over the distribution of $Y$s.)

(Actually, some more can be said under weak assumptions, but I bet (without checking) that most of this appears on the page(s) you are reading.)

Answer 2

For different values of $y$, we can define $\operatorname{Var}(X|Y=y)$ to be the variance of $X$ conditioned on the event $Y=y$. Now, let $\operatorname{Var}(X|Y)$ be a function that takes the value $\operatorname{Var}(X|Y=y)$ on any input $y$. Recall that a random variable is a function whose value is determined by random outcomes. Hence, $\operatorname{Var}(X|Y)$ can also be regarded as a random variable whose value is determined based on $y$. For instance, when $y$ is discrete we have: $$E[\operatorname{Var}(X|Y)] = \sum_y \operatorname{Var}(X|Y=y)\cdot \Pr(Y=y)$$