Category:Conditional Expectation Unchanged on Conditioning on Independent Sigma-Algebra
This category contains pages concerning Conditional Expectation Unchanged on Conditioning on Independent Sigma-Algebra:
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 $\map \sigma X$ be the $\sigma$-algebra generated by $X$.
Let $\HH \subseteq \Sigma$ be a sub-$\sigma$-algebra that is independent of $\map \sigma {\map \sigma X, \GG}$, the $\sigma$-algebra generated by $\map \sigma X \cup \GG$.
Let $\map \sigma {\GG, \HH}$ be the $\sigma$-algebra generated by $\GG \cup \HH$.
Let $\expect {X \mid \GG}$ be a version of the conditional expectation of $X$ given $\GG$.
Let $\expect {X \mid \map \sigma {\GG, \HH} }$ be a version of the conditional expectation of $X$ given $\map \sigma {\GG, \HH}$.
Then:
- $\expect {X \mid \map \sigma {\GG, \HH} } = \expect {X \mid \GG}$ almost surely.
Pages in category "Conditional Expectation Unchanged on Conditioning on Independent Sigma-Algebra"
The following 2 pages are in this category, out of 2 total.