Finding a function of $Pois(lambda_1)$ and $Pois(lambda_2)$ such that the distribution depends on $frac{lambda_1}{lambda_1+lambda_2}$ only.


Suppose that $Xsim Pois(lambda_1)$ and $Ysim Pois(lambda_2)$ are two independent Poisson random variables. Can we find a function $f(X,Y)$ of $X$ and $Y$, where $f(X,Y)$ does not involve $lambda_1$ and $lambda_2$, such that the distribution of $Zequiv f(X,Y)$ depends on the parameter $frac{lambda_1}{lambda_1+lambda_2}$ only?

I know there is a result that $X|X+Y=n$ follows a binomial distribution with parameters $n$ and $frac{lambda_1}{lambda_1+lambda_2}$. But can we find a statistic $Z= f(X,Y)$ without conditioning such that $Z$ depends on the parameter $frac{lambda_1}{lambda_1+lambda_2}$ only?

I tried some naive functions such as $f(X,Y)=frac{X}{X+Y+1}$ and $f(X,Y)=frac{X}{X^2+Y^2+1}$, but it turns out that those functions do not work.