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PQ220 6234F.Ch 03 13/04/2002 03:19 PM Page 60

60 CHAPTER 3 DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

3-1 DISCRETE RANDOM VARIABLES

Many physical systems can be modeled by the same or similar random experiments and ran-
dom variables. The distribution of the random variables involved in each of these common
systems can be analyzed, and the results of that analysis can be used in different applications
and examples. In this chapter, we present the analysis of several random experiments and
discrete random variables that frequently arise in applications. We often omit a discussion of
the underlying sample space of the random experiment and directly describe the distribution
of a particular random variable.

EXAMPLE 3-1 A voice communication system for a business contains 48 external lines. At a particular time,
the system is observed, and some of the lines are being used. Let the random variable X denote
the number of lines in use. Then, X can assume any of the integer values 0 through 48. When
the system is observed, if 10 lines are in use, x = 10.

EXAMPLE 3-2 In a semiconductor manufacturing process, two wafers from a lot are tested. Each wafer is
classified as pass or fail. Assume that the probability that a wafer passes the test is 0.8 and that
wafers are independent. The sample space for the experiment and associated probabilities are
shown in Table 3-1. For example, because of the independence, the probability of the outcome
that the first wafer tested passes and the second wafer tested fails, denoted as pf, is

P1pf 2 0.810.22 0.16

The random variable X is defined to be equal to the number of wafers that pass. The
last column of the table shows the values of X that are assigned to each outcome in the
experiment.

EXAMPLE 3-3 Define the random variable X to be the number of contamination particles on a wafer in semi-
conductor manufacturing. Although wafers possess a number of characteristics, the random
variable X summarizes the wafer only in terms of the number of particles.
The possible values of X are integers from zero up to some large value that represents the
maximum number of particles that can be found on one of the wafers. If this maximum num-
ber is very large, we might simply assume that the range of X is the set of integers from zero
to infinity.

Note that more than one random variable can be defined on a sample space. In Example
3-3, we might define the random variable Y to be the number of chips from a wafer that fail
the final test.

Table 3-1 Wafer Tests

Outcome
Wafer 1 Wafer 2 Probability x
Pass Pass 0.64 2
Fail Pass 0.16 1
Pass Fail 0.16 1
Fail Fail 0.04 0