I am curious whether two Poisson distributions with two different rate parameters may have the same probability value at some positive integer argument i-e I am trying to solve
$$\frac{e^{-\lambda _0} \lambda _0^x}{x!} \overset{?}{=} \frac{e^{-\lambda _1} \lambda _1^x}{x!} \quad \quad \text{where $ \: x \in \mathbb {Z}^+$ and $\lambda_0 \neq \lambda_1.$} $$
I did try to start with numerical methods but didn't reach any where. Any idea would be helpful.
Consider the function $f(\lambda)=e^{-\lambda}\lambda^n$. Its derivative is $e^{-\lambda}(n\lambda^{n-1}-\lambda^n)$, or equivalently $e^{-\lambda}\lambda^{n-1}(n-\lambda)$.
Take for example $n=1$. Then $f(\lambda)$ is increasing until $\lambda=1$, and then decreasing. So there are infinitely many pairs $(\lambda_0,\lambda_1)$ with $\lambda_0\lt 1$ and $\lambda_1\gt 1$ at which we have equality. A similar result holds for any other positive $n$.