Is there an intuitive way of viewing the Law of Total Expectation $\mathbb{E}\big[\mathbb{E}[X|Y]\big]=\mathbb{E}[X]?$

Question

Law of total expectation

If $\mathbb{E}\big[|X|\big]$ finite then for any $Y,\;\mathbb{E}\big[\mathbb{E}[X\mid Y]\big]=\mathbb{E}[X]$

I remember reading this for the first time and thinking... hold up, what?

The proof is simple, but I am wondering whether there is an intuitive reason why we might expect this result?

Answer 1

The conditional expectation of $X$ with respect to $Y$ is our best estimate of $X$ given exact knowledge of $Y.$ The expectation of any variable is our best estimate, given no specific knowledge about any variable at all. It seems reasonable then that our a priori expectation of the variable $E[X|Y]$ before we have any knowledge of $Y$ is just the general expectation of $X.$

Answer 2

That depends on why this is violating your intuition. For me, the basic intuition behind it is, $E(X)$ is the expected value of the random variable $X$, across all possible conditions. For any random variable $Y$, let $y_1, y_2, y_3, \ldots$ represent the possible values of $Y$. Then these $y_i$ are also, in some sense, a "cover set" of all possible conditions, and therefore if you do a weighted average of the conditional expected values $E(X \mid Y)$, you should obtain the overall expected value of $X, E(X)$.