Question 2.44: (Sums of Independent Binomial Random Variables) If X and Y a......
(Sums of Independent Binomial Random Variables) If X and Y are independent binomial random variables with parameters (n, p) and (m, p), respectively, then what is the distribution of X + Y?
The moment generating function of X + Y is given by
\phi_{X+Y}(t)=\phi_{X}(t)\phi_{Y}(t)=(p e^{t}+1-p)^{n}(p e^{t}+1-p)^{m}=(p e^{t}+1-p)^{m+n}
But (p e^{t}+(1-p))^{m+n} is just the moment generating function of a binomial random variable having parameters m + n and p. Thus, this must be the distribution of X + Y.
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