ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Alphabet
Deduction system
Theory
Zermelo-Fraenkel set theory
Set
Binary cartesian set product
Binary relation
Map
Simple map
Simple function
Measurable simple complex function
Simple integral
Unsigned basic integral
Unsigned basic expectation
Basic expectation
Random real number moment
Expectation
Conditional expectation representative
Conditional expectation
Definition D2012
Conditional probability
Formulation 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $\mathcal{G} \subseteq \mathcal{F}$ is a D470: Subsigma-algebra of $\mathcal{F}$ on $\Omega$
(ii) $E \in \mathcal{F}$ is an D1716: Event in $P$
The conditional probability of $E$ in $P$ given $\mathcal{G}$ is the D3161: Random real number \begin{equation} \mathbb{P}(E \mid \mathcal{G}) := \mathbb{E}(I_E \mid \mathcal{G}) \end{equation}
Formulation 1
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $\mathcal{G} \subseteq \mathcal{F}$ is a D470: Subsigma-algebra of $\mathcal{F}$ on $\Omega$
(ii) $E \in \mathcal{F}$ is an D1716: Event in $P$
The conditional probability of $E$ in $P$ given $\mathcal{G}$ is the D3161: Random real number \begin{equation} \Omega \to [0, 1], \quad \omega \mapsto \mathbb{E}(I_E \mid \mathcal{G})(\omega) \end{equation}
Children
Conditionally independent event collection
Results
R4932
R4931
R4930
Bayes' theorem in the case of two pullback events
Bayes' theorem in the case of event and complement
Bayes' theorem in the case of two events
Conditional probability given independent random variable
Conditional probability given independent sigma-algebra
Conditional probability of almost surely true event
Conditional probability of complement event
Conditional probability of the empty event
Conditional probability of the sample space
Expectation of conditional probability
Law of total probability for a countable partition of events of positive probability
Law of total probability for complement partition in terms of random variables
Law of total probability for complex expectation in terms of pullback events
Law of total probability for complex expectation in terms of pullback events of a discrete random variable
Probability calculus expression for probability conditioned on event of nonzero probability