Proof of the Law of Total Expectation

The law of total expectation (or the law of iterated expectations or the tower property) is

$\mathbb{E}[X] = \mathbb{E}[\mathbb{E}[X \mid Y]].$

There are proofs of the law of total expectation that require weaker assumptions. However, the following proof is straightforward for anyone with an elementary background in probability. Let $X$ and $Y$ are two random variables. Assume their densities exist. Then,

$\begin{aligned} \mathbb{E}[X] &= \int_{x} x f_X(x)\text{d}x \\ &= \int_{x} x \int_{y} f_{X,Y}(x, y) \text{d}x\text{d}y \\ &= \int_{x} x \int_{y} f_{X \mid Y}(x, y) f_Y(y)\text{d}y \\ &\stackrel{\star}{=} \int_{y} \Bigg[ \int_{x} x f_{X \mid Y}(x, y) \Bigg] f_Y(y)\text{d}y \\ &= \int_{y} \mathbb{E}[X \mid Y] f_Y(y)\text{d}y \\ &= \mathbb{E}[\mathbb{E}[X \mid Y]]. \end{aligned}$

Step $\star$ does not hold without a regularity condition because, in general, you cannot change the order of integration. Thus, we need Fubini’s theorem to hold, and we are done.