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February, 1979 Continous Versions of Regular Conditional Distributions
Sandy Zabell
Ann. Probab. 7(1): 159-165 (February, 1979). DOI: 10.1214/aop/1176995158


Let $X$ and $Y$ be random variables and assume $X$ has a density $f_X(x)$. An inversion theorem for the conditional expectation $E(Y\mid X = x)$ is derived which generalizes and simplifies that of Yeh. As an immediate corollary an almost-sure version of Bartlett's formula for the conditional characteristic function of $Y$ given $X = x$ is obtained. This result is applied to show the existence under regularity conditions of a version of the regular conditional distribution $P\{dy\mid X = x\}$ which is well defined for those values of $x$ such that $f_X(x) \neq 0$.


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Sandy Zabell. "Continous Versions of Regular Conditional Distributions." Ann. Probab. 7 (1) 159 - 165, February, 1979.


Published: February, 1979
First available in Project Euclid: 19 April 2007

zbMATH: 0392.60013
MathSciNet: MR515823
Digital Object Identifier: 10.1214/aop/1176995158

Primary: 60E05
Secondary: 60B15

Keywords: conditional characteristic function , conditional expectation , regular conditional distributions

Rights: Copyright © 1979 Institute of Mathematical Statistics

Vol.7 • No. 1 • February, 1979
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