Regular conditional probability: Difference between revisions

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Revision as of 18:00, 21 January 2014

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Consider a Radon space <math> \Omega </math> (that is a probability measure defined on a Radon space endowed with the Borel sigma-algebra) and a real-valued random variable ''T''. As discussed above, in this case there exists a regular conditional probability with respect to ''T''. Moreover we can alternatively define the '''regular conditional probability''' for an event ''A'' given a particular value ''t'' of the random variable ''T'' in the following manner:

:<math> P (A|T=t) = \lim_{U\nisupset \{T= t\}} \frac {P(A\cap U)}{P(U)},</math>

where the [[Limit (mathematics)|limit]] is taken over the [[Net (mathematics)|net]] of [[Open set|open]] [[Neighbourhood (mathematics)|neighborhoods]] ''U'' of ''t'' as they become [[Subset|smaller with respect to set inclusion]]. This limit is defined if and only if the probability space is [[Radon space|Radon]], and only in the support of ''T'', as described in the article. This is the restriction of the transition probability to the support of ''T''. To describe this limiting process rigorously: