Total Expectation Theorem

From ProofWiki
Jump to navigation Jump to search

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

\(\ds \sum_i \expect {X \mid B_i}\) \(=\) \(\ds \sum_i \sum_x x \, \map \Pr {\set {X = x} \cap B_i}\) Definition of Conditional Expectation
\(\ds \) \(=\) \(\ds \sum_x x \map \Pr {\set {X \in x} \cap \paren {\bigcup_i B_i} }\)
\(\ds \) \(=\) \(\ds \sum_x x \map \Pr {X = x}\)
\(\ds \) \(=\) \(\ds \expect X\) Definition of Expectation

$\blacksquare$


Also known as

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

Some sources give this as the law of total expectation.


Sources