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» Conditional expectation
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 basic 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 basic real number \begin{equation} \Omega \to [0, 1], \quad \omega \mapsto \mathbb{E}(I_E \mid \mathcal{G})(\omega) \end{equation}
Also known as
Posterior probability
Child definitions
» D2795: Conditionally independent event collection
Results
» R2556: Probability calculus expression for probability conditioned on event of nonzero probability
» 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
» R3640: Conditional probability given independent sigma-algebra
» R3641: Conditional probability given independent random variable
» R3649: Expectation of conditional probability
» R3404: Bayes' theorem in the case of two events
» R4340: Bayes' theorem in the case of event and complement
» R4341: Bayes' theorem in the case of two pullback events
» R4338: Conditional probability of almost surely true event
» R4339: Conditional probability of almost surely false event