Content deleted Content added
→Conditional discrete distributions: Lay out |
DonnabuffL (talk | contribs) m →Measure-theoretic formulation: Typo, condition on sigma algebra G, not B |
||
Line 72:
==Measure-theoretic formulation==
Let <math>(\Omega, \mathcal{F}, P)</math> be a probability space, <math>\mathcal{G} \subseteq \mathcal{F}</math> a <math>\sigma</math>-field in <math>\mathcal{F}</math>. Given <math>A\in \mathcal{F}</math>, the [[Radon-Nikodym theorem]] implies that there is<ref>[[#billingsley95|Billingsley (1995)]], p. 430</ref> a <math>\mathcal{G}</math>-measurable random variable <math>P(A\mid\mathcal{G}):\Omega\to \mathbb{R}</math>, called the '''conditional probability''', such that<math display="block">\int_G P(A\mid\mathcal{G})(\omega) dP(\omega)=P(A\cap G)</math>for every <math>G\in \mathcal{G}</math>, and such a random variable is uniquely defined up to sets of probability zero. A conditional probability is called [[Regular conditional probability|'''regular''']] if <math> \operatorname{P}(\cdot\mid\mathcal{
Special cases:
| |||