Expectation of Conditional Expectation
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Theorem
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $\GG \subseteq \Sigma$ be a sub-$\sigma$-algebra.
Let $X$ be a integrable random variable.
Let $\expect {X \mid \GG}$ be a version of the conditional expectation of $X$ given $\GG$.
Then:
- $\expect {\expect {X \mid \GG} } = \expect X$
Proof
We have:
\(\ds \expect {\expect {X \mid \GG} }\) | \(=\) | \(\ds \int_\Omega \expect {X \mid \GG} \rd \Pr\) | Definition of Expectation | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_\Omega X \rd \Pr\) | Definition of Conditional Expectation on Sigma-Algebra | |||||||||||
\(\ds \) | \(=\) | \(\ds \expect X\) | Definition of Expectation |
$\blacksquare$
Sources
- 1991: David Williams: Probability with Martingales ... (previous) ... (next): $9.7$: Properties of conditional expectation: a list