# 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.