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Given 2 random variables $X,Y$, is it possible to write conditional expectation $\mathbb{E}[X|Y]$ in terms of their joint distributional function $F_{X,Y}(x,y)$?

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2 Answers 2

Given $F_{X,Y}(x,y)$, you can obtain $P(x,y)$ and also $P(y)$ by integrate out $x$. This makes you know conditional probability $P(x|y)$, so does conditional expectation.

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what is $P(x,y)$ here? –  chandu1729 Oct 5 '13 at 15:28
    
Please note $X,Y$ can be discrete, continuous or have both components. –  chandu1729 Oct 5 '13 at 15:29
    
Yes, this is always true no matter what kind of probability. Integrate on continuous part and sum on discrete part –  Shuchang Oct 5 '13 at 15:32
    
Can you please define what is $P(x,y)$ and how how you will obtain it from $F_{X,Y}$ –  chandu1729 Oct 5 '13 at 15:35
    
To interval where $F_{X,Y}$ is continuous, $P(x,y)=\frac{\partial}{\partial x}\frac{\partial}{\partial x}F_{x,y}$ while discrete case replace derivative to minus –  Shuchang Oct 5 '13 at 16:51
up vote 0 down vote accepted

Yes, this could be done using Radon-Nikodym derivative

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