Applied Statistics and Probability for Engineers

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 Xdenote the random vari-

able that counts the number of flaws in a length of Lmillimeters of wire and suppose that the

average number of flaws in Lmillimeters is.

The probability distribution of Xcan be found by reasoning in a manner similar to the pre-

vious example. Partition the length of wire into nsubintervals 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 Xas approximately a bino-

mial random variable. Because

we obtain

That is, the probability that a subinterval contains a flaw is. With small enough subinter-

vals, nis very large and pis very small. Therefore, the distribution of Xis 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.

n

pn

E 1 X 2 np



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 Xthat equals the number of counts in the interval is a Poisson

random variable with parameter ,and the probability mass function of Xis

f 1 x 2  (3-15)

ex

x!

x0, 1, 2,p

0 

Definition

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