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Conditional density

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The density of a conditional distribution. Let be a probability space, let be the -algebra of Borel sets on the line, let be a sub--algebra of , let

be the conditional distribution of with respect to , and let

be the conditional distribution function of with respect to . If

then is called the conditional density of the distribution of with respect to the -algebra .

If and are random variables, is the density of the distribution of and is the joint density of the distribution of and , then

defines the conditional density of the distribution of the random variable for fixed values of for which .

References

[1] Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian)
How to Cite This Entry:
Conditional density. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conditional_density&oldid=46440
This article was adapted from an original article by V.G. Ushakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article