I'm stuck again... the question I'm trying to figure out is:
Let X be Poisson(2) be independent of Y which is Poisson(3) and let W=X+Y. Find the distribution of X conditional on W=10.
Now I have figured out that the sum of two independent random variables, which have Poisson distribution, also has a Poisson distribution, specifically of Poisson(a_1+a_2) where a_1, a_2 are the parameters of the two indep random variables (that was in the qn I did
preceeding this one).
Thus, I know W will have distribution Poisson(5). I also know what the probablility functions look like for X being Poisson(2) and W being Poisson(5) are.
Now in order to find the distribution of X conditional on W=10 I want to figure out P(X|W=10) ... I think this is what I want at least.
Now using the conditional probablility definition you have
P(X|W=10) = P(X,W=10) ---------- P(W=10)
I can easily figure out P(W=10) but my problem is that I don't know how to figure out P(X,W=10) ... What is this intersection equal to? Could somebody help please?