Category:Conditional Expectation
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This category contains results about Conditional Expectation.
Definitions specific to this category can be found in Definitions/Conditional Expectation.
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be a discrete random variable on $\struct {\Omega, \Sigma, \Pr}$.
Let $B$ be an event in $\struct {\Omega, \Sigma, \Pr}$ such that $\map \Pr B > 0$.
The conditional expectation of $X$ given $B$ is written $\expect {X \mid B}$ and defined as:
- $\expect {X \mid B} = \ds \sum_{x \mathop \in \image X} x \condprob {X = x} B$
where:
- $\condprob {X = x} B$ denotes the conditional probability that $X = x$ given $B$
whenever this sum converges absolutely.
Subcategories
This category has the following 4 subcategories, out of 4 total.
C
Pages in category "Conditional Expectation"
The following 21 pages are in this category, out of 21 total.
C
- Condition for Conditional Expectation to be Almost Surely Non-Negative
- Conditional Dominated Convergence Theorem
- Conditional Expectation Conditioned on Event of Non-Zero Probability
- Conditional Expectation Conditioned on Trivial Sigma-Algebra
- Conditional Expectation is Linear
- Conditional Expectation is Monotone
- Conditional Expectation of Constant
- Conditional Expectation of Measurable Random Variable
- Conditional Expectation of Non-Negative Random Variable is Non-Negative
- Conditional Expectation of Sum of Squared Increments of Square-Integrable Martingale
- Conditional Expectation Unchanged on Conditioning on Independent Sigma-Algebra
- Conditional Expectation Unchanged on Conditioning on Independent Sigma-Algebra/Corollary
- Conditional Fatou's Lemma
- Conditional Jensen's Inequality
- Conditional Monotone Convergence Theorem
- Conditional Reverse Fatou's Lemma