Q: Qual problem: One dim. sufficient statistic for $\lambda$ based on the data (X,Y) where X-Piosson$(\Lambda)$ and Y-Bernolli$(\lambda/(1+\lambda))$ Qual problem: 
We observe the pair $(X,Y)$  where $X$-Poisson$(\lambda)$ and $Y$-Bernoulli$(\lambda/(1+\lambda)),$  $\lambda$ is unknown.
Find one dimensional sufficient statistic for $\lambda$ based on (X,Y)
I thought that T(X,Y)=Y will be sufficient, or only taking the X values is sufficient but i am not sure. 
Thanks in advance, 

A: Presumably $X$ and $Y$ are independent.
Hint: the joint pmf for $(X,Y)$ is 
$$p_{X,Y}(x,y) = \dfrac{e^{-\lambda}}{1+\lambda} \dfrac{ \lambda^{x+y}}{x!}
\ (x \in \mathbb N, y \in \{0,1\})$$ 
Do you know the factorization criterion (Fisher-Neyman factorization theorem)?
