Qual problem: One dim. sufficient statistic for $\lambda$ based on the data (X,Y) where X-Piosson$(\Lambda)$ and Y-Bernolli$(\lambda/(1+\lambda))$

Question

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,

Answer 1

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)?