Applied Statistics and Probability for Engineers

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 variablesthat 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 Xdenote

the number of lines in use. Then, Xcan 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 passor 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

The random variable Xis defined to be equal to the number of wafers that pass. The

last column of the table shows the values of Xthat are assigned to each outcome in the

experiment.

EXAMPLE 3-3 Define the random variable Xto be the number of contamination particles on a wafer in semi-

conductor manufacturing. Although wafers possess a number of characteristics, the random

variable Xsummarizes the wafer only in terms of the number of particles.

The possible values of Xare 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 Xis 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 Yto be the number of chips from a wafer that fail

the final test.

P 1 pf 2 0.8 1 0.2 2 0.16

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

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