# Exponential family and geometric distribution: how do we prove the sum of independent geometric random variables has negative binomial distribution?

Let $$X$$ be the number of failures before the first success in a sequence of Bernoulli trials with probability of success $$\theta$$. Then $$P_{\theta}[X = k] = (1-\theta)^{k}\theta$$, $$k = 1,2,\ldots$$ This is called the geometric distribution ($$\mathcal{G}(\theta))$$.\

(a) Show that the family of geometric distributions is a one-parameter exponential family with $$T(x) = x$$.

(b) Deduce that if $$X_{1},X_{2},\ldots, X_{n}$$ is a sample from $$\mathcal{G}(\theta)$$, then the distributions of $$\sum_{i=1}^{n}X_{i}$$ form a one-parameter exponential family.

(c) Show that $$\sum_{i=1}^{n}X_{i}$$ in part (b) has a negative binomial distribution with parameters $$(n,\theta)$$ defined by \begin{align*} P_{\theta}\left[\sum_{i=}^{n}X_{i} = k\right] = {{n+k-1}\choose{k}}(1-\theta)^{k}\theta^{n} \end{align*}

where $$k = 0,1,2,\ldots$$

MY ATTEMPT

So far, I am fine with the first. As to the one-parameter exponential family, the author considers the following expression \begin{align*} p(x,\theta) = h(x)\exp(\eta(\theta)T(x) - B(\theta)) \end{align*}

Could someone help me with the second and third questions? Thanks in advance!

• there are many ways to show this result, one way that is somewhat simple to do (if you assume each trial is independent) is to write out the generating/characteristic functions for each trial. Multiplication of these should bring you to the corresponding negative binomial generating/characteristic function. – Lucas Roberts May 13 at 22:49
• The approach suggested by @Lucas is a fine one. An intuitively appealing one refers to the contexts in which these distributions typically arise: namely, the number of independent trials needed to observe $k$ failures obviously equals the sum of the numbers of trials needed to observe the first failure, then the second, ..., and finally the $k^\text{th}.$ – whuber May 13 at 23:15