Page 200 - Fundamentals of Probability and Statistics for Engineers

Some Important Discrete Distributions 183

k
e
p k†ˆ ; k ˆ 0; 1; ... : 6:54†
X
k!
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 the probability 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

100 0 100 0 100
p 0†ˆ 0:015† 0:985† ˆ 0:985† ˆ 0:2206:
X
0
Since n is large and p is small in this case, the Poisson approximation is
appropriate and we obtain
0 1:5
1:5† e
p 0†ˆ ˆ e 1:5 ˆ 0:223;
X
0!
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 ˆ 1000, and p ˆ 1/500 ˆ 0.002, and the Poisson
approximation is appropriate. Using Equation (6.54), we have ˆ np ˆ 2,
and the desired probability is
3 3 2
2
2 e e 2
0.18.
p 3 †ˆ ˆ 0:18:
X
3! 3
The examples above demonstrate that the Poisson distribution finds applica-
tions in problems where the probability of an event occurring is small. For this
reason, it is often referred to as the distribution of rare events.

6.4 SUMMARY

We have introduced in this chapter several discrete distributions that are used
extensively in science and engineering. Table 6.3 summarizes some of the
important properties associated with these distributions, for easy reference.

TLFeBOOK