Suppose $X \sim Binomial (Y,\delta)$. For the random variable $X$ can I compute the following conditional probability as follows? $P(X=x|Y\leq y)=$ $\sum\limits_{k=1}^{y} P(X=x|Y=k)$, where Y is a discrete random variable having the value of positive integers.
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1What is $P(X|E)$ for a random variable $X$ and an event $E$? – user10354138 Oct 25 '18 at 22:50
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@user10354138 Thanks for the question. I revised the question. – user3509199 Oct 26 '18 at 02:25
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Yes, you can, provided that the summands are disjoint, i.e. that $$ \left( {X \cap Y = k} \right) \cap \left( {X \cap Y = j} \right) = \emptyset \quad \left| {\;k \ne j} \right. $$
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