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Let there be three random variables $X$, $Y$ and $Z$.

How can I prove the folowing?

$P(X|Y) = \sum\limits_{z} P(X,z|Y)$

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    How is $P(X|Y)$ defined for random variables $X$ and $Y$? (I am not familiar with that notation, and it is of course not the same as $P(A|B)$ for measurable sets $A$ and $B$.)2012-03-21

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A correct formulation would be that $\mathrm P(X=x\mid Y=y)=\sum\limits_z\mathrm P(X=x,Z=z\mid Y=y)$ for every $y$ such that $\mathrm P(Y=y)\ne0$.

This formula is an example of the fact that $\mathrm P(A\mid C)=\sum\limits_k\mathrm P(A\cap B_k\mid C)$ for every events $A$, $(B_k)_k$ and $C$ such that $\mathrm P(C)\ne0$ and such that $(B_k)_k$ is a partition of the underlying probability space. In turn, this fact follows from the observation that the events $A\cap B_k\cap C$ are disjoint and that their union over $k$ is $A\cap C$.