$X$ and $Y$ are independent random variables taking non-negative value.
Suppose $Z=X+Y$ and $P(Z=n)\gt0\quad(n=0,1,2…)$
I need to prove two conditions below are equivalent:
(1) There are $\lambda_a\gt0, \lambda\gt0$
$$P(X=x)=\frac{e^{-\lambda_a}\lambda^x_a}{x!}, \enspace P(Y=y)=\frac{e^{-\lambda_b}\lambda^y_b}{y!},\quad x,y=0,1,2,…$$
(2) There is $p(0\lt p\lt 1)$ and $n$ is a non-negative value.
$$P(X=x\mid Z=n)=\left( \begin{array}{c} n \\ x \\ \end{array} \right)p^x(1-p)^{n-x}, \quad x=0,1,…n$$
I have tried to prove for (2)->(1), I did not make it as I did not have PDF of $Z$