Given an arbitrary joint PMF $p_{X,Y}(x, y)$, we know that the expectation of some $g(x)$ or $X$ and $Y$ is:

$\displaystyle{\mathbf E[g(X,Y)] = \sum\limits_{x}\sum\limits_{y} g(x,y)\cdot p_{X,Y}(x,y)}$

If the formula of the conditional expectation of some $f(X,Y\mid Y=y)$ as follow?

$\displaystyle{\mathbf E[f(X,Y\mid Y = y)] = \sum\limits_{x}\sum\limits_{y} f(x,y)\cdot p_{X,Y\,\mid\, Y=y}(x,y\,\mid\,Y=y)}$


  • $\begingroup$ Found something similar, but not sure how to relate $\displaystyle{\mathbf E[Z\mid Y = y] = \sum\limits_{z\in\,\textrm{range}\{Z\mid Y = y\}} z\cdot p_{Z\,\mid\, Y = y}(z\,\mid\,Y = y)}$. $\endgroup$ – muxo Feb 14 at 13:23

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