Page 112 - Applied Statistics And Probability For Engineers

PQ220 6234F.Ch 03 13/04/2002 03:19 PM Page 90

90 CHAPTER 3 DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

EXAMPLE 3-31 Flaws occur at random along the length of a thin copper wire. Let X denote the random vari-
able that counts the number of flaws in a length of L millimeters of wire and suppose that the
average number of flaws in L millimeters is .
The probability distribution of X can be found by reasoning in a manner similar to the pre-
vious example. Partition the length of wire into n subintervals of small length, say, 1 microm-
eter each. If the subinterval chosen is small enough, the probability that more than one flaw
occurs in the subinterval is negligible. Furthermore, we can interpret the assumption that
flaws occur at random to imply that every subinterval has the same probability of containing
a flaw, say, p. Finally, if we assume that the probability that a subinterval contains a flaw is in-
dependent of other subintervals, we can model the distribution of X as approximately a bino-
mial random variable. Because

E1X2 np

we obtain

p n

That is, the probability that a subinterval contains a flaw is n . With small enough subinter-
vals, n is very large and p is very small. Therefore, the distribution of X is obtained as in the
previous example.

Example 3-31 can be generalized to include a broad array of random experiments. The
interval that was partitioned was a length of wire. However, the same reasoning can be
applied to any interval, including an interval of time, an area, or a volume. For example,
counts of (1) particles of contamination in semiconductor manufacturing, (2) flaws in rolls
of textiles, (3) calls to a telephone exchange, (4) power outages, and (5) atomic particles
emitted from a specimen have all been successfully modeled by the probability mass func-
tion in the following definition.

Definition
Given an interval of real numbers, assume counts occur at random throughout the in-
terval. If the interval can be partitioned into subintervals of small enough length such
that

(1) the probability of more than one count in a subinterval is zero,
(2) the probability of one count in a subinterval is the same for all subintervals
and proportional to the length of the subinterval, and
(3) the count in each subinterval is independent of other subintervals, the ran-
dom experiment is called a Poisson process.
The random variable X that equals the number of counts in the interval is a Poisson
random variable with parameter 0 , and the probability mass function of X is

e x
f 1x2 x 0, 1, 2, p (3-15)
x!