So the problem goes like this X~Poisson(lambda) and Y|X=k ~ Bi(k,p).

Prove Y~Poisson(lambda*p)

I already got to show that if Y~Poisson(mu) then X|X+Y=k ~ Bi(k, lamda/(lambda+mu))

So I to start the problem I thought

P(Y=y)=sum( P(Y=y|X=k)*P(X=k)) for k>=0

Eventually I get something like this

P(Y=y)= (p/1-p)^y * e^(-lamda)/y! * sum ( (lamda-p)^k / (k-n)!)

I tried to move on and I can't. After long while trying I thought of solving the problem through moment generator functions. But I'm very lost because I got a some of multpilications, is it true that Mt(Y) = sum(Mt(Y|X=k)*Mt(X) ?

Thanks for reading and any hint