Let \(X_{1}, X_{2}, \ldots, X_{r}\) be \(r\) independent random variables each having the same geometric distribution.

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

\[\left[p e^{t} /\left(1-(1-p) e^{t}\right)\right]^{r}\]

(b) Relate the sum to the total number of trials to obtain \(r\) successes. This distribution, is given by

\[\left(\begin{array}{l}x-1 \-1\end{array}\right) p^{r} q^{x-r}, x=r, r+1, \cdots\]

(see page 135)

(c) Obtain the first two moments of this negative binomial by differentiating the mgf.

Transcribed Image Text:

Next, suppose the probability that the accident occurs on any interval of length L is L/200, with L measured in miles. Note that this arbitrary assignment of prob- ability is consistent with Axioms 1 and 2 on page 70, since the probabilities are all nonnegative and less than or equal to 1, and P(S) = 200 = 1. 200 So far, we are considering only events represented by intervals which form part of the line segment from 0 to 200. Using Axiom 3' on page 120, we can also obtain probabilities of events that are not intervals but which can be represented by the union of finitely many or countably many intervals. Thus, for two nonoverlapping intervals of length L1 and L2 we have a probability of 41 +42 200 and for an infinite sequence of nonoverlapping intervals of length L1, L2, L3, ..., we have a probability of L1+2+3+... 200