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

[Not a Statistician]

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]

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 \( Z = z \), you need to know/list all possible pairs of \( (X, Y) \) such that \( X + Y = z \).

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

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