Suppose that $X\sim Pois(\lambda_1)$ and $Y\sim Pois(\lambda_2)$ 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 $Z\equiv f(X,Y)$ depends on the parameter $\frac{\lambda_1}{\lambda_2}$ only?
I tried some naive functions such as $f(X,Y)=\frac{X}{Y+1}$ and $f(X,Y)=\frac{X^2}{Y^2+1}$, but it turns out that those functions do not work.