Let be a probability space, the -algebra of Borel sets on the line, a random variable defined on and a sub--algebra of . A function defined on is called a (regular) conditional distribution of the random variable with respect to the -algebra if:
a) for fixed the function is -measurable;
b) with probability one, for fixed the function is a probability measure on ;
c) for arbitrary ,
Similarly one can define the conditional distribution of a random element with values in an arbitrary measurable space . If is a complete separable metric space and is the -algebra of Borel sets, then the conditional distribution of the random element relative to any -algebra , , exists.
The function is called the conditional distribution function of the random variable with respect to the -algebra .
The conditional distribution (conditional distribution function) of a random variable with respect to a random variable is defined as the conditional distribution (conditional distribution function) of with respect to the -algebra generated by .
The conditional distribution function of a random variable with respect to is a Borel function of ; for its value is called the conditional distribution function of for a fixed value of . If has a density , then
where is the joint distribution function of and .
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Conditional distribution. V.G. Ushakov (originator), Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conditional_distribution&oldid=11344