Prove conditional probability relation of random variables
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Let there be three random variables $X$, $Y$ and $Z$.
How can I prove the folowing?
$P(X|Y) = \sum\limits_{z} P(X,z|Y)$
probabilityprobability-theory
asked 2012-03-20
user id:27228
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This is nothing more than an application of the definition of conditional probability. – 2012-03-20
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Do you mean this? $P(b | a) = \frac{P(a, b)}{P(a)}$ ? – 2012-03-20
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Yep, and now remember that P(A)=sum_x P(A, x), or more generally that P(A)= sum_i (A, B_i) where B_i are pairwise disjoint and $\cup_i B_i = \Omega$ ($\Omega$ is the whole space). – 2012-03-20
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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