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==Conditional discrete distributions==
For [[discrete random variable]]s, the conditional [[probability mass function]] of <math>Y</math> given <math>X=x</math> can be written according to its definition as:
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==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{{sfnp|Billingsley|1995|p=430}} 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{G})(\omega) </math> is a [[probability measure]] on <math>(\Omega, \mathcal{F})</math> for all <math>\omega \in \Omega</math> a.e.
Special cases:
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