Random Variable of two other random variables. Confusion to the max!

#1
Here is the question at hand:
I don't even know how to begin this question. I'm not looking for a complete solution per se but I don't even know where to begin.
Suppose that X is a binomial random variable with parameters (m, θ) and Y is another
binomial random variable with parameters (n, θ) and suppose that X and Y are independent.
Let Z = X+Y . Show that Z is another Binomial random variable with parameters (m+n, θ).
Hint:
P
Note that you have to find the PMF of Z: P(Z = z). You may use the fact that
sum over k of (n choose k)(n choose (z-k) = (m plus n) choose z where the sum ranges over k for which the binomial coefficients are
defined.
 

BGM

TS Contributor
#2
The technique in a general mathematical context is called convolution.

In probability context, you can understand it as law of total probability.

The first key idea is when [math] Z = z [/math], you need to know/list all possible pairs of [math] (X, Y) [/math] such that [math] X + Y = z [/math].

For example, when [math] Z = 2 [/math], the possible pairs are [math] (0, 2), (1, 1), (2, 0) [/math], provided that [math] m, n > 1 [/math].

Try to generalize a bit and see if you can apply the law of total probability to solve that.