Where n is a positive integer, 0 < p < 1 , q = 1 - p.
How would I show the mgf of X is
how would I derive the mgf of Y?
The mgf of a discrete distribution is :
So here, we have :
Now just recall Newton's binomial formula :
As for the second one, if the Xi's are independent, then the mgf of Y is the product of the mgf of the Xi.
If they're identically distributed, this will just be
The point of using MGF's is that you hope to recognize the resulting one.
Here you added m INDEPENDENT binomials.
This techniques proves that the result is a binomial with same probability of success (p) and now the number of trials is nm.
The MGF of X+Y where they are indep bin(n,p) and bin(m,p), SAME p, different sample sizes is
proving that X+Y is bin(n+m,p)
So, for you the mean is (nm)p and the variance is (nm)pq.