First of all you should know the sum of two indep Poissons is another Poisson and it's mean is the sum of the other two rv's means.
Then
where . You should be able to finish this.
I'll check your work, if you post it.
Put in the poisson work and reduce it.
Hit quote on my work above and continue with that ratio of probabilities.
You'll learn TEX as you do the math.
It's a win-win for you.
My solution:
P(X=k|x+y=n) is bionominal distribution with BIN( n, lambda 1/ (lambda 1+lambda 2) )
so E(x|x+y) = n* lambda 1/ (lambda 1+lambda 2)
Am I right? Um.. I don't know where can I click "thank"? coz this is the first time I use this website.....Where is it?