---
tag: 04RRB6HM00
kind: statement
role: theorem
name: Law of total expectation
aliases: [tower-property, law-of-iterated-expectations, adam-law]
msc: [60A10]
uses: [0YAMJRK04E, 0DCT0QZVQ1, 0E5G5FMT2R]
sources:
  - url: https://en.wikipedia.org/wiki/Law_of_total_expectation
    locator: Proof in the general case
same_as:
  wikidata: Q2247110
  wikipedia: 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|0YAMJRK04E]].
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|0E5G5FMT2R]].
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 $0.6 \cdot 70 + 0.4 \cdot 80 = 74$.