The sum of independent NegativeBinomial variables with the same beta parameter is distributed as which?

• A. Negative Binomial(sum r, beta)
• B. Poisson(lambda sum)
• C. Binomial(sum r, p)
• D. Gamma(sum r, beta)

Answer: (A)

The sum of independent Negative Binomial variables that share the same beta parameter is itself Negative Binomial, with the total r equal to the sum of the individual r’s.

This works because a Negative Binomial NB(r, beta) can be viewed through its probability generating function as G(s) = [(1 − beta)/(1 − beta s)]^r. For independent X1, X2, ..., Xn with parameters (r1, beta), (r2, beta), ..., the PGF of the sum is the product of their PGFs:

G_sum(s) = ∏ [(1 − beta)/(1 − beta s)]^{ri} = [(1 − beta)/(1 − beta s)]^{sum ri},

which matches the PGF of NB(sum ri, beta). Equivalently, you can think of NB as a Poisson-Gamma mixture or as a sum of ri geometric variables with the same success probability; summing across independent NB variables accumulates the counts of those geometric components, yielding NB(sum ri, beta).

So the combined distribution is Negative Binomial with parameters sum of the r’s and the same beta.