Law of total expectation

Statement

Let $X$ be an integrable real random variable on $(\Omega, \mathcal{F}, P)$ and let $\mathcal{G} \subseteq \mathcal{F}$ be a σ-algebra. Then $$E\left[E[X \mid \mathcal{G}]\right] = E[X].$$ In particular, $E[E[X \mid Y]] = E[X]$ for each random variable $Y$ on the same space.

Remarks

Proof: put $G = \Omega$ in the defining property of the conditional expectation. For a partition of $\Omega$ into events $A_1, A_2, \dots$ with $P(A_i) > 0$, it gives $E[X] = \sum_i E[X \mid A_i] \, P(A_i)$. This extends the law of total probability. More generally, for σ-algebras $\mathcal{H} \subseteq \mathcal{G}$, $E[E[X \mid \mathcal{G}] \mid \mathcal{H}] = E[X \mid \mathcal{H}]$. This is the tower property.

Explanations

L1 human

To find an overall average, you can first average inside each group and then average the group averages, with the group sizes as weights. If $60\%$ of the students have mean score $70$ and $40\%$ have mean score $80$, the overall mean is $0.6 \cdot 70 + 0.4 \cdot 80 = 74$.