Suppose $X$ and $Y$ are independent non-negative integer random variables, such that $X+Y$ has binomial distribution. Prove that $X$ and $Y$ are also binomial.
I tried to just write the obvious equation, but it seems that this approach does not take me anywhere... Is there an easy way to see this result, without any tediuos computations?
We know that $Z=X+Y$ is a Binomial $B(n,p)$
Because $X,Y$ are non-negative indepedent integer random variables, they must be restricted to $0\le X,Y\le n$.
Now, let $G_X(t)$ and $G_Y(t)$ be the GF of $X$ and $Y$.
Because of the above $G_X(t)$, $G_Y(t)$ are polinomial of degree at most $n$. And $G_Z(t)$ is a polynomial of degree $n$:
$$G_Z(t)= (1 -p +pz )^n = G_X(t)G_Y(t)$$
Because the polynomial is already factorized, (and we have the restriction $G(1)=1$) there must exist some $ 0 \le n_x \le n$ such that
$$ G_X(t)= (1 -p +pz )^{n_x}$$ $$ G_Y(t)= (1 -p +pz )^{n_y}$$
with $n_y = n- n_x$.
Then both $X,Y$ are Binomials with same parameter $p$ as $Z$.