Fundamentals of Probability and Statistics for Engineers

volumes, and if it is reasonable to assume that the probability of finding k flaws in
any region depends only on the volume and not on the shape of the region.
Other physical situations in which the Poisson distribution is used include
bacteria counts on a Petri plate, the distribution of airplane-spread fertilizers
in a field, and the distribution of industrial pollutants in a given region.

Example 6.14.A good example of this application is the study carried out by
Clark (1946) concerning the distribution of flying-bomb hits in one part of London
during World War 2. The area is divided into 576 small areas of 0 .25km^2 each.
In Table 6.2, the number of areas with exactly k hits is recorded and
is compared with the predicted number based on a Poisson distribution, with
number of total hits per number of areas 537/576 0 .932. We see an
excellent agreement between the predicted and observed results.

6.3.2 The Poisson Approximation to the Binomial Distribution

Let X be a random variable having the binomial distribution with

Consider the case when 0, in such a way that remains
fixed. Wenotethat isthemean ofX, which isassumed to remain constant. Then,

As , the factorials n! and (n k)! appearing in the binomial coefficient
can be approximated by using the Stirling’s formula [Equation (4.78)]. We also
note that

U sing these relationships in Equation (6.52) then gives, after some manipulation,

Table 6. 2 Comparison of the observed and theoretical

distributionsofflying-bombhits,forExample 6. 14

012345

229 211 93 35 7 1

226. 7 211. 4 98. 5 30. 6 7. 1 1. 6

182 FundamentalsofProbabilityandStatisticsforEngineers

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