Page 186 - Fundamentals of Probability and Statistics for Engineers

Some Important Discrete Distributions 169

which may seem much smaller than what we experience in similar situa-
tions.
Example 6.6. Problem: assume that the probability of a specimen failing
during a given experiment is 0.1. What is the probability that it will take more
than three specimens to have one surviving the experiment?
Answer: let X denote the number of trials required for the first specimen
to survive. It then has a geometric distribution with p ˆ 0.9. The desired
probability is

3
3
P X > 3†ˆ 1 F X 3†ˆ 1 1 q †ˆ 0:1† ˆ 0:001:
Example 6.7. Problem: let the probability of occurrence of a flood of magni-
tude greater than a critical magnitude in any given year be 0.01. Assuming that
floods occur independently, determine EfNg , the average return period. The
average return period, or simply return period, is defined as the average number
of years between floods for which the magnitude is greater than the critical
magnitude.
Answer: it is clear that N is a random variable with a geometric distribution
and p ˆ 0: 01. The return period is then

1
EfNgˆ ˆ 100 years:
p
The critical magnitude which gives rise to EfNgˆ 100 years is often referred to
as the ‘100-year flood’.

6.1.3 NEGATIVE BINOMIAL DISTRIBUTION

A natural generalization of the geometric distribution is the distribution of
random variable X representing the number of Bernoulli trials necessary for the
rth success to occur, where r is a given positive integer.
In order to determine p (k) for this case, let A be the event that the first k 1
X
trials yield exactly r 1 successes, regardless of their order, and B the event that
a success turns up at the kth trial. Then, owing to independence,
p k† ˆ P A \ B† ˆ P A†P B†: 6:18†
X

Now, P(A) obeys a binomial distribution with parameters k 1 and r 1, or

k 1
P A†ˆ p r 1 k r ; k ˆ r; r ‡ 1; ... ; 6:19†
q
r 1

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