Refer to Exercise 6.40. Let \(X_{1}, X_{2}, \ldots, X_{n}\) be \(n\) independent random variables each having a negative binomial distribution with success probability \(p\) but where \(X_{i}\) has parameter \(r_{i}\).

(a) Show that the \(\operatorname{mgf} M_{\sum X_{i}}(t)=E\left(e^{t\left(X_{1}+X_{2}+\cdots+X_{r}\right)}\right)\) of the \(\operatorname{sum} \sum X_{i}\) is

\[\left[p e^{t} /\left(1-(1-p) e^{t}\right)\right]^{\sum_{i=0}^{n} r_{i}}\]

(b) Identify the form of this mgf and specify the distribution of \(\sum X_{i}\).

Data From Exercise 6.40

Transcribed Image Text:

6.40 Let X1, X2,..., X, be r independent random variables each having the same geometric distribution. (a) Show that the moment generating function Mx(t) = E(e(x1+x2+...+X+)) of the sum is [pe/(1 (1p) e')]' (b) Relate the sum to the total number of trials to ob- tain r successes. This distribution, is given by x- ( += }) pq', x = rr + 1,... (c) Obtain the first two moments of this negative bi- nomial by differentiating the mgf.