# Sum of binomial distributed random variables

Let $X \sim Bi(n,p), Y \sim Bi(m,p)$. “Visual arguments” suggest that $X+Y \sim Bi(m+n,p)$. However, I am unable to prove that.

Using the definition I can reduce the problem to $$\sum_{i=0}^k \binom{m}{i} \binom{n}{k-i} = \binom{n+m}{k},$$ which is – according to Wolfram Alpha – correct. But I have no idea for a proof here as well. (I only tried to use the definition of the binomial coefficients, but that did not help me.)

So: How to prove the statement above?

• 31 minutes. 
– Did
Jul 16 '14 at 16:25
• This assumes that X and Y are independent.
– Did
Jul 16 '14 at 16:26

A combinatorial interpretation of the identity follows: Suppose you have $n$ boys and $m$ girls. The RHS counts the number of ways to choose $k$ children out of the total $n+m$ children regardless of sex; the LHS counts the number of ways to choose $i$ girls and $k-i$ boys for each $i = 0, 1, 2, \ldots, k$. Obviously, the two are equal.