Fundamentals of Probability and Statistics for Engineers

This Poisson approximation to the binomial distribution can be used to advan-
tage from the point of view of computational labor. It also establishes the fact
that a close relationship exists between these two important distributions.

Example 6.15.Problem: suppose that the probability of a transistor manu-
factured by a certain firm being defective is 0.015. What is theprobability that
there is no defective transistor in a batch of 100?
Answer: let X be the number of defective transistors in 100. The desired
probability is

Since n is large and p is small in this case, the Poisson approximation is
appropriate and we obtain

which is very close to the exact answer. In practice, the Poisson approximation
is frequently used when n > 10, and p < 0:1.

Example 6.16.Problem: in oil exploration, the probability of an oil strike
in the North Sea is 1 in 500 drillings. What is the probability of having exactly
3 oil-producing wells in 1000 explorations?
Answer: in this case, n 1000, and p 1/500 0 .002, and the Poisson
approximation is appropriate. Using Equation (6.54), we have
and the desired probability is

3

23 e^2

3!

0. 18.

TheexamplesabovedemonstratethatthePoissondistributionfindsapplica-
tionsinproblemswheretheprobabilityofaneventoccurringissmall.Forthis
reason, it is often referred to as the distribution of rare events.

6.4 Summary

Wehaveintroducedinthischapterseveraldiscretedistributionsthatareused
extensively in science and engineering. Table 6. 3 summarizes some of the
importantpropertiesassociatedwiththesedistributions,foreasyreference.

SomeImportantDiscreteDistributions 183

pX…k†ˆ

ke

k!

; kˆ 0 ; 1 ;...: … 6 : 54 †

pX… 0 †ˆ

100

0



… 0 : 015 †^0 … 0 : 985 †^100 ^0 ˆ… 0 : 985 †^100 ˆ 0 : 2206 :

pX… 0 †ˆ

… 1 : 5 †^0 e^1 :^5

0!

ˆe^1 :^5 ˆ 0 : 223 ;

ˆ ˆ ˆ

ˆnpˆ2,

pX…†ˆ

23 e^2

3

ˆ 0 : 18 :