# Total Expectation Theorem

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## Contents

## Theorem

Let $\mathcal E = \struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $x$ be a discrete random variable on $\mathcal E$.

Let $\set {B_1 \mid B_2 \mid \cdots}$ be a partition of $\omega$ such that $\map \Pr {B_i} > 0$ for each $i$.

Then:

- $\displaystyle \expect X = \sum_i \expect {X \mid B_i} \, \map \Pr {B_i}$

whenever this sum converges absolutely.

In the above:

- $\expect X$ denotes the expectation of $X$
- $\expect {X \mid B_i}$ denotes the conditional expectation of $X$ given $B_i$.

## Proof

\(\displaystyle \sum_i \expect {X \mid B_i}\) | \(=\) | \(\displaystyle \sum_i \sum_x x \, \map \Pr {\set {X = x} \cap B_i}\) | Definition of Conditional Expectation | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \sum_x x \map \Pr {\set {X \in x} \cap \paren {\bigcup_i B_i} }\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \sum_x x \map \Pr {X = x}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \expect X\) | Definition of Expectation |

$\blacksquare$

## Also known as

Some sources refer to this as the **partition theorem**, which causes ambiguous as that name is used for other things as well.

## Sources

- 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $\S 2.5$: Conditional expectation and the partition theorem: Theorem $2 \ \text C$