Let there be three random variables $X$, $Y$ and $Z$.
How can I prove the folowing?
$P(X|Y) = \sum\limits_{z} P(X,z|Y)$
Let there be three random variables $X$, $Y$ and $Z$.
How can I prove the folowing?
$P(X|Y) = \sum\limits_{z} P(X,z|Y)$
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 $A\cap C$.