Fundamentals of Probability and Statistics for Engineers

Example 6.9.The negative binomial distribution is widely used in waiting-
time problems. Consider, for example, a car waiting on a ramp to merge into
freeway traffic. Suppose that it is 5th in line to merge and that the gaps between
cars on the freeway are such that there is a probability of 0.4 that they are
large enough for merging. Then, if X is the waiting time before merging for
this particular vehicle measured in terms of number of freeway gaps, it has
a negative binomial distribution with 5 and 0 .4. The mean waiting time
is, as seen from Equation (6.27),

6.2 M ultinomial D istribution

Bernoulli trials can be generalized in several directions. A useful generalization
is to relax the requirement that there be only two possible outcomes for each
trial. Let there be r possible outcomes for each trial, denoted by E 1 ,E 2 ,...,Er,
and let and A typical outcome of
n trials is a succession of symbols such as:

If we let random variable represent the number of Ei in a
sequen ce of n trials, the joint probability mass function (jpmf) of X 1 ,X 2 ,...,Xr,
is given by

where 0,1,2,..., 1,2,...,r,and

ProofforEquation6.30:wewanttoshowthatthecoefficientinEquation
(6. 30 ) is equal to the number of ways of placing k 1 letters E 1 ,k 2 letters E 2 ,...,
and kr letters Er in n boxes. This can be easily verified by writing

The first binomial coefficient is the number of ways of placing k 1 letters E 1
in n boxes; the second is the number of ways of placing k 2 letters E 2 in the
remaining unoccupiedboxes;andsoon.

172 FundamentalsofProbabilityandStatisticsforEngineers

rˆ pˆ

EfXgˆ

5

0 : 4

ˆ 12 : 5 gaps:

P(Ei)ˆpi,iˆ1,...,r, p 1 ‡p 2 ‡‡prˆ1.

E 2 E 1 E 3 E 3 E 6 E 2 ...:

Xi,iˆ1,2,...,r,

pX 1 X 2 ...Xr…k 1 ;k 2 ;...;kr†ˆ

n!

k 1 !k 2 !...kr!

pk 11 pk 22 ...pkrr; … 6 : 30 †

kjˆ jˆ k 1 ‡k 2 ‡‡krˆn.

n!

k 1 !k 2 !...kr!

ˆ

n

k 1



nk 1

k 2





nk 1 k 2 kr 1

kr



:

nk 1