Page 114 - Applied Statistics And Probability For Engineers

PQ220 6234F.Ch 03 13/04/2002 03:19 PM Page 92

92 CHAPTER 3 DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

EXAMPLE 3-32 For the case of the thin copper wire, suppose that the number of flaws follows a Poisson dis-
tribution with a mean of 2.3 flaws per millimeter. Determine the probability of exactly 2 flaws
in 1 millimeter of wire.
Let X denote the number of flaws in 1 millimeter of wire. Then, E(X) 2.3 flaws and

e 2.3 2.3 2
P1X 22 0.265
2!

Determine the probability of 10 flaws in 5 millimeters of wire. Let X denote the number
of flaws in 5 millimeters of wire. Then, X has a Poisson distribution with

E1X2 5 mm
2.3 flaws/mm 11.5 flaws

Therefore,
11.5 10
11.5
P1X 102 e 0.113
10!
Determine the probability of at least 1 flaw in 2 millimeters of wire. Let X denote the
number of flaws in 2 millimeters of wire. Then, X has a Poisson distribution with

E1X2 2 mm
2.3 flaws/mm 4.6 flaws

Therefore,

P1X 12 1 P1X 02 1 e 4.6 0.9899

EXAMPLE 3-33 Contamination is a problem in the manufacture of optical storage disks. The number of particles
of contamination that occur on an optical disk has a Poisson distribution, and the average number
of particles per centimeter squared of media surface is 0.1. The area of a disk under study is 100
squared centimeters. Find the probability that 12 particles occur in the area of a disk under study.
Let X denote the number of particles in the area of a disk under study. Because the mean
number of particles is 0.1 particles per cm 2

2
2
E1X2 100 cm
0.1 particles/cm 10 particles
Therefore,
e 10 10 12
P1X 122 0.095
12!
The probability that zero particles occur in the area of the disk under study is

P1X 02 e 10 4.54
10 5

Determine the probability that 12 or fewer particles occur in the area of the disk under
study. The probability is

12 e 10 10 i
# # #
P1X 122 P1X 02 P1X 12 P1X 122 a
i 0 i!