The theoretical question asks:
Suppose that $X$ and $Y$ are independent geometric random variables with the same parameter $P$. Find the value and verify:
$P{(X=i| X+Y = n )}$?
My initial reaction is that since they are independent and have the same parameter P, that the given is not necessary and the answer would just end up being $ (1-P)^{(i-1)} P$ but I am pretty sure that I am missing something, can someone help me get started on this proof?
Best Answer
No, the problem is that $X$ and $Y$ satisfy $X,Y\geq 1$. If you know that $X+Y=4$ the value of $X$ can't be greater than $3$.
Hint: $$P(X=i~|~X+Y=n)=\frac{P(Y=n-i,X=i)}{P(X+Y=n)}$$ What do you know about the distribution of $X+Y$?