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
D2795: Conditionally independent event collection
Results
R4932
R4931
R4930
R4341: Bayes' theorem in the case of two pullback events
R4340: Bayes' theorem in the case of event and complement
R3404: Bayes' theorem in the case of two events
R3641: Conditional probability given independent random variable
R3640: Conditional probability given independent sigma-algebra
R4338: Conditional probability of almost surely true event
R4720: Conditional probability of complement event
R4823: Conditional probability of the empty event
R4822: Conditional probability of the sample space
R3649: Expectation of conditional probability
R2677: Law of total probability for a countable partition of events of positive probability
R4802: Law of total probability for complement partition in terms of random variables
R3783: Law of total probability for complex expectation in terms of pullback events
R3784: Law of total probability for complex expectation in terms of pullback events of a discrete random variable
R2556: Probability calculus expression for probability conditioned on event of nonzero probability