Handbook of Industrial Automation - Richard L. Shell and Ernest L. Hall Part 1 doc

Selected subjects from both under-graduate- and graduate-level topics from industrial, electrical, computer, and mechanical engineering as well asmaterial science are included to provide

Marcel Dekker, Inc New York•Basel

Handbook of

Industrial Automation

edited by

Richard L Shell Ernest L Hall

University of Cincinnati Cincinnati, Ohio

ISBN: 0-8247-0373-1

This book is printed on acid-free paper

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270 Madison Avenue, New York, NY 10016

Copyright # 2000 by Marcel Dekker, Inc All Rights Reserved

Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical,including photocopying, micro®lming, and recording, or by any information storage and retrieval system, without permission inwriting from the publisher

Current printing (last digit):

10 9 8 7 6 5 4 3 2 1

PRINTED IN THE UNITED STATES OF AMERICA

This handbook is designed as a comprehensive reference for the industrial automation engineer Whether in a small

or large manufacturing plant, the industrial or manufacturing engineer is usually responsible for using the latest andbest technology in the safest, most economic manner to build products This responsibility requires an enormousknowledge base that, because of changing technology, can never be considered complete The handbook willprovide a handy starting reference covering technical, economic, certain legal standards, and guidelines that should

be the ®rst source for solutions to many problems The book will also be useful to students in the ®eld as it provides

a single source for information on industrial automation

The handbook is also designed to present a related and connected survey of engineering methods useful in avariety of industrial and factory automation applications Each chapter is arranged to permit review of an entiresubject, with illustrations to provide guideposts for the more complex topics Numerous references are provided toother material for more detailed study

The mathematical de®nitions, concepts, equations, principles, and application notes for the practicing industrialautomation engineer have been carefully selected to provide broad coverage Selected subjects from both under-graduate- and graduate-level topics from industrial, electrical, computer, and mechanical engineering as well asmaterial science are included to provide continuity and depth on a variety of topics found useful in our work inteaching thousands of engineers who work in the factory environment The topics are presented in a tutorial style,without detailed proofs, in order to incorporate a large number of topics in a single volume

The handbook is organized into ten parts Each part contains several chapters on important selected topics Part

1 is devoted to the foundations of mathematical and numerical analysis The rational thought process developed inthe study of mathematics is vital in developing the ability to satisfy every concern in a manufacturing process.Chapters include: an introduction to probability theory, sets and relations, linear algebra, calculus, differentialequations, Boolean algebra and algebraic structures and applications Part 2 provides background information onmeasurements and control engineering Unless we measure we cannot control any process The chapter topicsinclude: an introduction to measurements and control instrumentation, digital motion control, and in-processmeasurement

Part 3 provides background on automatic control Using feedback control in which a desired output is compared

to a measured output is essential in automated manufacturing Chapter topics include distributed control systems,stability, digital signal processing and sampled-data systems Part 4 introduces modeling and operations research.Given a criterion or goal such as maximizing pro®t, using an overall model to determine the optimal solutionsubject to a variety of constraints is the essence of operations research If an optimal goal cannot be obtained, thencontinually improving the process is necessary Chapter topics include: regression, simulation and analysis ofmanufacturing systems, Petri nets, and decision analysis

iii

Part 5 deals with sensor systems Sensors are used to provide the basic measurements necessary to control amanufacturing operation Human senses are often used but modern systems include important physical sensors.Chapter topics include: sensors for touch, force, and torque, fundamentals of machine vision, low-cost machinevision and three-dimensional vision Part 6 introduces the topic of manufacturing Advanced manufacturing pro-cesses are continually improved in a search for faster and cheaper ways to produce parts Chapter topics include: thefuture of manufacturing, manufacturing systems, intelligent manufacturing systems in industrial automation, mea-surements, intelligent industrial robots, industrial materials science, forming and shaping processes, and moldingprocesses Part 7 deals with material handling and storage systems Material handling is often considered a neces-sary evil in manufacturing but an ef®cient material handling system may also be the key to success Topics include

an introduction to material handling and storage systems, automated storage and retrieval systems, tion, and robotic palletizing of ®xed- and variable-size parcels

containeriza-Part 8 deals with safety and risk assessment Safety is vitally important, and government programs monitor themanufacturing process to ensure the safety of the public Chapter topics include: investigative programs, govern-ment regulation and OSHA, and standards Part 9 introduces ergonomics Even with advanced automation,humans are a vital part of the manufacturing process Reducing risks to their safety and health is especiallyimportant Topics include: human interface with automation, workstation design, and physical-strength assessment

in ergonomics Part 10 deals with economic analysis Returns on investment are a driver to manufacturing systems.Chapter topics include: engineering economy and manufacturing cost recovery and estimating systems

We believe that this handbook will give the reader an opportunity to quickly and thoroughly scan the ®eld ofindustrial automation in suf®cient depth to provide both specialized knowledge and a broad background of speci®cinformation required for industrial automation Great care was taken to ensure the completeness and topicalimportance of each chapter

We are grateful to the many authors, reviewers, readers, and support staff who helped to improve the script We earnestly solicit comments and suggestions for future improvements

manu-Richard L ShellErnest L Hall

Preface iii

Contributors ix

Part 1 Mathematics and Numerical Analysis

1.1 Some Probability Concepts for Engineers 1

Enrique Castillo and Ali S Hadi

1.2 Introduction to Sets and Relations

Part 2 Measurements and Computer Control

2.1 Measurement and Control Instrumentation Error-Modeled Performance

Patrick H Garrett

2.2 Fundamentals of Digital Motion Control

Ernest L Hall, Krishnamohan Kola, and Ming Cao

v

2.3 In-Process Measurement

William E Barkman

Part 3 Automatic Control

3.1 Distributed Control Systems

Dobrivoje Popovic

3.2 Stability

Allen R Stubberud and Stephen C Stubberud

3.3 Digital Signal Processing

Richard Brook and Denny Meyer

4.2 A Brief Introduction to Linear and Dynamic Programming

Part 5 Sensor Systems

5.1 Sensors: Touch, Force, and Torque

Richard M Crowder

5.2 Machine Vision Fundamentals

Prasanthi Guda, Jin Cao, Jeannine Gailey, and Ernest L Hall

6.2 Manufacturing Systems

Jon Marvel and Ken Bloemer

6.3 Intelligent Manufacturing in Industrial Automation

George N Saridis

6.4 Measurements

John Mandel

6.5 Intelligent Industrial Robots

Wanek Golnazarian and Ernest L Hall

6.6 Industrial Materials Science and Engineering

Part 7 Material Handling and Storage

7.1 Material Handling and Storage Systems

William Wrennall and Herbert R Tuttle

7.2 Automated Storage and Retrieval Systems

Stephen L Parsley

7.3 Containerization

A Kader Mazouz and C P Han

7.4 Robotic Palletizing of Fixed- and Variable-Size/Content Parcels

Hyder Nihal Agha, William H DeCamp, Richard L Shell, and Ernest L Hall

Part 8 Safety, Risk Assessment, and Standards

9.1 Perspectives on Designing Human Interfaces for Automated Systems

Anil Mital and Arunkumar Pennathur

9.2 Workstation Design

Christin Shoaf and Ashraf M Genaidy

9.3 Physical Strength Assessment in Ergonomics

Sean Gallagher, J Steven Moore, Terrence J Stobbe, James D McGlothlin, and Amit Bhattacharya

Part 10 Economic Analysis

10.1 Engineering Economy

Thomas R Huston

10.2 Manufacturing-Cost Recovery and Estimating Systems

Eric M Malstrom and Terry R Collins

Index 863

viii Contents

William H DeCamp Motoman, Inc., West Carrollton, Ohio

Steve Dickerson Department of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GeorgiaVerna Fitzsimmons Department of Mechanical, Industrial, and Nuclear Engineering, University of Cincinnati,Cincinnati, Ohio

Jeannine Gailey Department of Mechanical, Industrial, and Nuclear Engineering, University of Cincinnati,Cincinnati, Ohio

Sean Gallagher Pittsburgh Research Laboratory, National Institute for Occupational Safety and Health,Pittsburgh, Pennsylvania

Patrick H Garrett Department of Electrical and Computer Engineering and Computer Science, University ofCincinnati, Cincinnati, Ohio

Ashraf M Genaidy Department of Mechanical, Industrial, and Nuclear Engineering, University of Cincinnati,Cincinnati, Ohio

Wanek Golnazarian General Dynamics Armament Systems, Burlington, Vermont

Prasanthi Guda Department of Mechanical, Industrial, and Nuclear Engineering, University of Cincinnati,Cincinnati, Ohio

Ali S Hadi Department of Statistical Sciences, Cornell University, Ithaca, New York

Ernest L Hall Department of Mechanical, Industrial, and Nuclear Engineering, University of Cincinnati,Cincinnati, Ohio

C P Han Department of Mechanical Engineering, Florida Atlantic University, Boca Raton, Florida

Thomas R Huston Department of Mechanical, Industrial, and Nuclear Engineering, University of Cincinnati,Cincinnati, Ohio

Avraam I Isayev Department of Polymer Engineering, The University of Akron, Akron, Ohio

Ki Hang Kim Mathematics Research Group, Alabama State University, Montgomery, Alabama

Krishnamohan Kola Department of Mechanical, Industrial, and Nuclear Engineering, University of Cincinnati,Cincinnati, Ohio

Eric M Malstromy Department of Industrial Engineering, University of Arkansas, Fayetteville, ArkansasJohn Mandel National Institute of Standards and Technology, Gaithersburg, Maryland

Jon Marvel Padnos School of Engineering, Grand Valley State University, Grand Rapids, Michigan

A Kader Mazouz Department of Mechanical Engineering, Florida Atlantic University, Boca Raton, FloridaJames D McGlothlin Purdue University, West Lafayette, Indiana

M Eugene Merchant Institute of Advanced Manufacturing Sciences, Cincinnati, Ohio

Denny Meyer Institute of Information and Mathematical Sciences, Massey University±Albany, PalmerstonNorth, New Zealand

Angelo B Mingarelli School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, CanadaAnil Mital Department of Industrial Engineering, University of Cincinnati, Cincinnati, Ohio

J Steven Moore Department of Occupational and Environmental Medicine, The University of Texas HealthCenter, Tyler, Texas

x Contributors

*Retired

yDeceased

Diego A Murio Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio

Lawrence E Murr Department of Metallurgical and Materials Engineering, The University of Texas at El Paso, ElPaso, Texas

Joseph H Nurre School of Electrical Engineering and Computer Science, Ohio University, Athens, OhioStephen L Parsley ESKAYCorporation, Salt Lake City, Utah

Arunkumar Pennathur University of Texas at El Paso, El Paso, Texas

Dobrivoje Popovic Institute of Automation Technology, University of Bremen, Bremen, Germany

Shivakumar Raman Department of Industrial Engineering, University of Oklahoma, Norman, OklahomaGeorge N Saridis Professor Emeritus, Electrical, Computer, and Systems Engineering Department, RensselaerPolytechnic Institute, Troy, New York

Richard L Shell Department of Mechanical, Industrial, and Nuclear Engineering, University of Cincinnati,Cincinnati, Ohio

Christin Shoaf Department of Mechanical, Industrial, and Nuclear Engineering, University of Cincinnati,Cincinnati, Ohio

J B Srivastava Department of Mathematics, Indian Institute of Technology, Delhi, New Delhi, India

Terrence J Stobbe Industrial Engineering Department, West Virginia University, Morgantown, West VirginiaAllen R Stubberud Department of Electrical and Computer Engineering, University of California Irvine, Irvine,California

Stephen C Stubberud ORINCON Corporation, San Diego, California

Hiroyuki Tamura Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka, Japan

Fred J Taylor Department of Electrical and Computer Engineering and Department of Computer andInformation Science Engineering, University of Florida, Gainesville, Florida

Herbert R Tuttle Graduate Engineering Management, University of Kansas, Lawrence, Kansas

William Wrennall The Leawood Group Ltd., Leawood, Kansas

Contributors xi

Many engineering applications involve some element

of uncertainty [1] Probability is one of the most

com-monly used ways to measure and deal with

uncer-tainty In this chapter we present some of the most

important probability concepts used in engineering

applications

The chapter is organized as follows Section 1.2 ®rst

introduces some elementary concepts, such as random

experiments, types of events, and sample spaces Then

it introduces the axioms of probability and some of the

most important properties derived from them, as well

as the concepts of conditional probability and

indepen-dence It also includes the product rule, the total

prob-ability theorem, and Bayes' theorem

Section 1.3 deals with unidimensional random

vari-ables and introduces three types of varivari-ables (discrete,

continuous, and mixed) and the corresponding

prob-ability mass, density, and distribution functions

Sections 1.4 and 1.5 describe the most commonly

used univariate discrete and continuous models,

respectively

Section 1.6 extends the above concepts of univariate

models to the case of bivariate and multivariate

mod-els Special attention is given to joint, marginal, and

conditional probability distributions

Section 1.7 discusses some characteristics of randomvariables, such as the moment-generating function andthe characteristic function

Section 1.8 treats the techniques of variable formations, that is, how to obtain the probaiblity dis-tribution function of a set of transformed variableswhen the probability distribution function of the initialset of variables is known Section 1.9 uses the transfor-mation techniques of Sec 1.8 to simulate univariateand multivariate data

trans-Section 1.10 is devoted to order statistics, givingmethods for obtaining the joint distribution of anysubset of order statistics It also deals with the problem

of limit or asymptotic distribution of maxima andminima

Finally, Sec 1.11 introduces probability plots andhow to build and use them in making inferences fromdata

1.2 BASIC PROBABILITY CONCEPTS

In this section we introduce some basic probabilityconcepts and de®nitions These are easily understoodfrom examples Classic examples include whether amachine will malfunction at least once during the

®rst month of operation, whether a given structurewill last for the next 20 years, or whether a ¯ood will

1

occur during the next year, etc Other examples include

how many cars will cross a given intersection during a

given rush hour, how long we will have to wait for a

certain event to occur, how much stress level a given

structure can withstand, etc We start our exposition

with some de®nitions in the following subsection

1.2.1 Random Experiment and Sample Space

Each of the above examples can be described as a

ran-dom experiment because we cannot predict in advance

the outcome at the end of the experiment This leads to

the following de®nition:

De®nition 1 Random Experiment and Sample

Space: Any activity that will result in one and only

one of several well-de®ned outcomes, but does not

allow us to tell in advance which one will occur is called

a random experiment Each of these possible outcomes is

called an elementary event The set of all possible

ele-mentary events of a given random experiment is called

Therefore, for each random experiment there is an

associated sample space The following are examples of

random experiments and their associated sample

spaces:

Rolling a six-sided fair die once yields

Waiting for a machine to malfunction yields

How many cars will cross a given intersection yields

De®nition 2 Union and Intersection: If C is a set

con-taining all elementary events found in A or in Bor in

both, then write C ˆ …A [ B† to denote the union of A

and B, whereas, if C is a set containing all elementary

events found in both A and B, then we write C ˆ …A \ B†

to denote the intersection of A and B

Referring to the six-sided die, for example, if

A ˆ f1; 3; 5g, B ˆ f2; 4; 6g, and C ˆ f1; 2; 3g, then …A [

f1; 3g and …A \ B† ˆ , where  denotes the empty set

Random events in a sample space associated with a

random experiment can be classi®ed into several types:

which contains more than one elementary event

is called a composite event Thus, for example,

observing an odd number when rolling a sided die once is a composite event because itconsists of three elementary events

six-2 Compatible vs mutually exclusive events Twoevents A and B are said to be compatible ifthey can simultaneously occur, otherwise theyare said to be mutually exclusive or incompatibleevents For example, referring to rolling a six-sided die once, the events A ˆ f1; 3; 5g and B ˆf2; 4; 6g are incompatible because if one eventoccurs, the other does not, whereas the events

A and C ˆ f1; 2; 3g are compatible because if weobserve 1 or 3, then both A and C occur

3 Collectively exhaustive events If the union ofseveral events is the sample space, then theevents are said to be collectively exhaustive.f1; 3; 5g and B ˆ f2; 4; 6g are collectivelyexhaustive events but A ˆ f1; 3; 5g and C ˆ f1;2; 3g are not

Then A and B are said to be complementaryevents or B is the complement of A (or viceversa) The complement of A is usually denoted

by A For example, in the six-sided die example,

if A ˆ f1; 2g, A ˆ f3; 4; 5; 6g Note that an eventand its complement are always de®ned with

A and A are always mutually exclusive and lectively exhaustive events, hence …A \ A† ˆ 1.2.2 Probability Measure

col-To measure uncertainty we start with a given sampletively exhaustive outcomes of a given experiment areare closed under the union, intersection, complemen-tary and limit operations Such a class is called a -algebra Then, the aim is to assign to every subset in 

a real value measuring the degree of uncertainty aboutits occurrence In order to obtain measures with clearphysical and practical meanings, some general andintuitive properties are used to de®ne a class of mea-sures known as probability measures

De®nition 3 Probability Measure: A function p ping any subset A   into the interval ‰0; 1Š is called aprobability measure if it satis®es the following axioms:

map-2 Castillo and Hadi

Axiom 2 Additivity: For any (possibly in®nite)

sequence, A1; A2; ; of disjoint subsets of , then

p[AiˆXp…Ai†

Axiom 1 states that despite our degree of uncertainty,

gation formula that can be used to compute the

prob-ability of a union of disjoint subsets It states that the

uncertainty of a given subset is the sum of the

uncer-tainties of its disjoint parts

From the above axioms, many interesting properties

of the probability measure can be derived For

example:

Property 1 Boundary: p…† ˆ 0

Property 2 Monotonicity: If A  B  , then

p…A†  p…B†

Property 3 Continuity±Consistency: For every

increasing sequence A1 A2 or decreasing

sequence A1 A2 of subsets of  we have

lim

i!1p…Ai† ˆ p… lim

i!1Ai†Property 4 Inclusion±Exclusion: Given any pair of

subsets A and B of , the following equality

always holds:

p…A [ B† ˆ p…A† ‡ p…B† p…A \ B† …1†

Property 1 states that the evidence associated with a

complete lack of information is de®ned to be zero

Property 2 shows that the evidence of the membership

of an element in a set must be at least as great as the

evidence that the element belongs to any of its subsets

In other words, the certainty of an element belonging

to a given set A must not decrease with the addition of

elements to A

Property 3 can be viewed as a consistency or a

con-tinuity property If we choose two sequences

conver-ging to the same subset of , we must get the same limit

of uncertainty Property 4 states that the probabilities

of the sets A; B; A \ B, and A [ B are not independent;

they are related by Eq (1)

Note that these properties respond to the intuitive

notion of probability that makes the mathematical

model valid for dealing with uncertainty Thus, for

example, the fact that probabilities cannot be larger

than one is not an axiom but a consequence of

Axioms 1 and 2

De®nition 4 Conditional Probability: Let A and Bbetwo subsets of variables such that p…B† > 0 Then, theconditional probability distribution (CPD) of A given B

is given by

p…A j B† ˆp…A \ B†p…B† …2†Equation (2) implies that the probability of A \ B can

be written as

p…A \ B† ˆ p…B†p…A j B† …3†This can be generalized to several events as follows:

p…A j B1; ; Bk† ˆp…A; Bp…B 1; ; Bk†

1; ; Bk† …4†1.2.3 Dependence and Independence

De®ntion 5 Independence of Two Events: Let A and B

be two events Then A is said to be independent of Bifand only if

p…A j B† ˆ p…A† …5†otherwise A is said to be dependent on B

Equation (5) means that if A is independent of B,then our knowledge of B does not affect our knowl-edge about A, that is, B has no information about A.Also, if A is independent of B, we can then combineEqs (2) and (5) and obtain

p…A \ B† ˆ p…A† p…B† …6†Equation (6) indicates that if A is independent of B,then the probability of A \ B is equal to the product oftheir probabilities Actually, Eq (6) provides a de®ni-tion of independence equivalent to that in Eq (5).One important property of the independence rela-tion is its symmetry, that is, if A is independent of B,then B is independent of A This is because

p…B j A† ˆp…A \ B†p…A† ˆp…A† p…B†p…A† ˆ p…B†

Because of the symmetry property, we say that A and

B are independent or mutually independent The cal implication of symmetry is that if knowledge of B isrelevant (irrelevant) to A, then knowledge of A is rele-vant (irrelevant) to B

practi-The concepts of dependence and independence oftwo events can be extended to the case of more thantwo events as follows:

Some Probability Concepts for Engineers 3

De®nition 6 Independence of a Set of Events: The

events A1; ; Am are said to be independent if and

only if

p…A1\ \ Am† ˆYm

iˆ1

p…Ai† …7†

otherwise they are said to be dependent

In other words, fA1; ; Amg are said to be

indepen-dent if and only if their intersection probability is equal

to the product of their individual probabilities Note

that Eq (7) is a generalization of Eq (6)

An important implication of independence is that it

is not worthwhile gathering information about

inde-pendent (irrelevant) events That is, independence

means irrelevance

From Eq (3) we get

p…A1\ A2† ˆ p…A1j A2† p…A2† ˆ p…A2j A1† p…A1†

This property can be generalized, leading to the

so-called product or chain rule:

p…A1\ \ An† ˆ p…A1† p…A2j A1†

p…An j A1\ \ An 1†1.2.4 Total Probability Theorem

Theorem 1 Total Probability Theorem: Let fA1; ;

Ang be a class of events which are mutually incompatible

and such that [

Theorem 2 Bayes' Theorem: Let fA1; ; Ang be a

class of events which are mutually incompatible and

1.3 UNIDIMENSIONAL RANDOMVARIABLES

In this section we de®ne random variables, distinguishamong three of their types, and present various ways ofpresenting their probability distributions

De®nition 7 Random Variable: A possible

vector-n, which assigns to each

ele-X…!† ˆ x, is called an n-dimensional random variable.random variable X is also known as the support of X.When n ˆ 1 in De®nition 7, the random variable issaid to be unidimensional and when n > 1, it is said

to be multidimensional In this and Secs 1.4 and 1.5,

we deal with unidimensional random variables.Multidimensional random variables are treated inSec 1.6

Example 1 Suppose we roll two dice once Let A bethe outcome of the ®rst die and Bbe the outcome of theconsists of 36 possible pairs (A,B), as shown in Fig 2.Suppose we de®ne a random variable X ˆ A ‡ B, that

is, X is the sum of the two numbers observed when we rolltwo dice once Then X is a unidimensional random vari-able The support of this random variable is the set f2;3; ; 12g consisting of 11 elements This is also shown

in Fig 2

1.3.1 Types of Random VariablesRandom variables can be classi®ed into three types:discrete, continuous, and mixed We de®ne and giveexamples of each type below

4 Castillo and Hadi

Figure 1 Graphical illustration of the total probability rule

De®nition 8 Discrete Random Variables: A random

variable is said to be discrete if it can take a ®nite or

countable set of real values

As an example of a discrete random variable, let X

denote the outcome of rolling a six-sided die once

Since the support of this random variable is the ®nite

set f1; 2; 3; 4; 5; 6g, then X is discrete random variable

The random variable X ˆ A ‡ B in Fig 2 is another

example of discrete random variables

De®nition 9 Continuous Random Variables: A

ran-dom variable is said to be continuous if it can take an

uncountable set of real values

For example, let X denote the weight of an object,

then X is a continuous random variable because it can

take values in the set fx : x > 0g, which is an

uncoun-table set

De®nition 10 Mixed Random Variables: A random

variable is said to be mixed if it can take an uncountable

set of values and the probability of at least one value of x

is positive

Mixed random variables are encountered often inengineering applications which involve some type ofcensoring Consider, for example, a life-testing situa-tion where n machines are put to work for a givenperiod of time, say 30 days Let Xi denotes the time

at which the ith machine malfunctions Then Xi is

a random variable which can take the values

fx : 0 < x  30g This is clearly an uncountable set.But at the end of the 30-day period some machinesmay still be functioning For each of these machinesall what we know is that Xi 30g Then the probabilitythat Xiˆ 30 is positive Hence the random variable Xi

is of the mixed type The data in this example is known

Some Probability Concepts for Engineers 5

Figure 2 Graphical illustration of an experiment consisting of rolling two dice once and an associated random variable which isde®ned as the sum of the two numbers observed

sored Of course, there are situations where both right

and left censoring are present

1.3.2 Probability Distributions of Random

Variables

So far we have de®ned random variables and their

support In this section we are interested in measuring

the probability of each of these values and/or the

prob-ability of a subset of these values We know from

In other words, we are interested in ®nding the

prob-ability distribution of a given random variable Three

equivalent ways of representing the probability

distri-butions of these random variables are: tables, graphs,

and mathematical functions (also known as

mathema-tical models)

1.3.3 Probability Distribution Tables

As an example of a probability distribution that can be

displayed in a table let us ¯ip a fair coin twice and let X

be the number of heads observed Then the sample

HT; HHg, where TH, for example, denotes the

out-come: ®rst coin turned up a tail and second a head

The sample space of the random variable X is then

f0; 1; 2g For example, X ˆ 0 occurs when we observe

TT The probability of each of these possible values of

X is found simply by counting how many elements of

We can see that X ˆ 0 occurs when we observe the

outcome TT, X ˆ 1 occurs when we observe either

HT or TH, and X ˆ 2 occurs when we observe HH

Since there are four equally likely elementary events in

p…X ˆ 0† ˆ 1=4, p…X ˆ 1† ˆ 2=4, and p…X ˆ 2† ˆ 1=4

This probability distribution of X can be displayed in a

table as in Table 1 For obvious reasons, such tables

are called probability distribution tables Note that to

denote the random variable itself we use an uppercaseletter (e.g., X), but for its realizations we use the cor-responding lowercase letter (e.g., x)

Obviously, it is possible to use tables to display theprobability distributions of only discrete random vari-ables For continuous random variables, we have touse one of the other two means: graphs or mathema-tical functions Even in discrete random variables withlarge number of elements in their support, tables arenot the most ef®cient way of displaying the probabilitydistribution

1.3.4 Graphical Representation of ProbabilitiesThe probability distribution of a random variable canequivalently be represented graphically by displayingvalues in the support of X on a horizontal line anderecting a vertical line or bar on top of each of thesevalues The height of each line or bar represents theprobability of the corresponding value of X Forexample, Fig 3 shows the probability distribution ofthe random variable X de®ned in Example 1

For continuous random variables, we have in®nitelymany possible values in their support, each of whichhas a probability equal to zero To avoid this dif®culty,

we represent the probability of a subset of values by anarea under a curve (known as the probability densitycurve) instead of heights of vertical lines on top of each

of the values in the subset

For example, let X represent a number drawn domly from the interval ‰0; 10Š The probability distri-bution of X can be displayed graphically as in Fig 4.The area under the curve on top of the support of Xhas to equal 1 because it represents the total probabil-ity Since all values of X are equally likely, the curve is

ran-a horizontran-al line with height equran-al to 1/10 The height

of 1/10 will make the total area under the curve equal

to 1 This type of random variable is called a

contin-6 Castillo and Hadi

Table 1 The Probability Distribution of the Random

Variable X De®ned as the Number of Heads Resulting

from Flipping a Fair Coin Twice

012

0.250.500.25 Figure 3 Graphical representation of the probability distri-bution of the random variable X in Example 1.

F…x† ˆ p…X  x† ˆ

…x

1f …x† dxNote that the cdfs are denoted by the uppercase

letters P…x† and F…x† to distinguish them from the

pmf p…x† and the pdf f …x† Note also that since p…X

ˆ x† ˆ 0 for the continuous case, then

p…X  x† ˆ p…X < x† The cdf has the following

properties as a direct consequence of the de®nitions

Every distribution function can be written as a

lin-ear convex combination of continuous

distribu-tions and step funcdistribu-tions

1.3.7 Moments of Random Variables

The pmf or pdf of random variables contains all the

information about the random variables For example,

given the pmf or the pdf of a given random variable,

we can ®nd the mean, the variance, and other moments

of the random variable The results in this section are

presented for the continuous random variables using

the pdf and cdf, f …x† and F…x†, respectively For the

discrete random variables, the results are obtained by

replacing f …x†, F…x†, and the integration symbol by

p…x†, P…x†, and the summation symbol, respectively

De®nition 11 Moments of Order k: Let X be a

ran-dom variable with pdf f …x†, cdf F…x†, and support A

Then the kth moment mkaround a 2 A is the real

Note that the Stieltjes±Lebesgue integral, Eq (10),

does not always exist In such a case we say that the

corresponding moment does not exist However, Eq

(10) implies the existence of

Ajx ajkf …x† dxwhich leads to the following theorem:

Theorem 3 Existence of Moments of Lower Order: Ifthe tth moment around a of a random variable X exists,then the sth moment around a also exists for 0 < s  t.The ®rst central moment is called the mean or theexpected value of the random variable X, and isdenoted by  or E‰XŠ Let X and Y be random vari-ables, then the expectation operator has the followingimportant properties:

E‰cŠ ˆ c, where c is a constant

E‰aX ‡ bY ‡ cŠ ˆ aE‰XŠ ‡ bE‰YŠ ‡ c; 8a; b; c 2 A

a  Y  b ) a  E‰YŠ  b:

jE‰YŠj  E‰j yjŠ:

The second moment around the mean is called thevariance of the random variable, and is denoted byVar…X† or 2 The square root of the variance, , iscalled the standard deviation of the random variable.The physical meanings of the mean and the varianceare similar to the center of gravity and the moment ofinertia, used in mechanics They are the central anddispersion measures, respectively

Using the above properties we can write

E‰…X a†2Š ˆ 2‡ … a†2

1.4 UNIVARIATE DISCRETE MODELS

In this section we present several important discreteprobability distributions that often arise in engineeringapplications.Table 2shows the pmf of these distribu-tions For additional probability distributions, seeChristensen [2] and Johnson et al [3]

8 Castillo and Hadi

1.4.3 The Binomial Distribution

Suppose now that we repeat a Bernoulli experiment n

times under identical conditions (that is, the outcome

of one trial does not affect the outcomes of the others)

In this case the trials are said to be independent

Suppose also that the probability of success is p and

that we are interested in the number of trials, X in

which the outcomes are successes The random

vari-able giving the number of successes after n realizations

of independent Bernoulli experiments is called a

bino-mial random variable and is denoted as B…n; p† Its pmf

is given inTable 2 Figure 6 shows some examples of

pmfs associated with binomial random variables

In certain situations the event X ˆ 0 cannot occur

The pmf of the binomial distribution can be modi®ed

to accommodate this case The resultant random able is called the nonzero binomial Its pmf is given inTable 2

vari-1.4.4 The Geometric or Pascal DistributionSuppose again that we repeat a Bernoulli experiment ntimes, but now we are interested in the random vari-able X, de®ned to be the number of Bernoulli trialsthat are required until we get the ®rst success Notethat if the ®rst success occurs in the trial number x,then the ®rst …x 1† trials must be failures (see Fig 7).Since the probability of a success is p and the prob-ability of the …x 1† failures is …1 p†x 1 (becausethe trials are independent), then the

p…X ˆ x† ˆ p…1 p†x 1 This random variable is calledthe geometric or Pascal random variable and isdenoted by G…p†

1.4.5 The Negative Binomial DistributionThe geometric distribution arises when we are inter-ested in the number of Bernoulli trials that are requireduntil we get the ®rst success Now suppose that wede®ne the random variable X as the number ofBernoulli trials that are required until we get the rthsuccess For the rth success to occur at the xth trial, wemust have …r 1† successes in the …x 1† previoustrials and one success in the rth trial (see Fig 8).This random variable is called the negative binomialrandom variable and is denoted by NB…r; p† Its pmf

is given in Table 2 Note that the gometric distribution

is a special case of the negative binomial distributionobtained by setting …r ˆ 1†, that is, G…p† ˆ NB…1; p†.1.4.6 The Hypergeometric Distribution

Consider a set of N items (products, machines, etc.), Ditems of which are defective and the remaining …N D†items are acceptable Obtaining a random sample ofsize n from this ®nite population is equivalent to with-drawing the items one by one without replacement

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Figure 5 A graph of the pmf and cdf of a Bernoulli

distribution

Figure 6 Examples of the pmf of binomial random variables Figure 7 Illustration of the Pascal or geometric randomvariable, where s denotes success and f denotes failure

1.5.3 The Gamma Distribution

Let Y be a Poisson random variable with parameter 

Let X be the time up to the kth Poisson event, that is,

the time it takes for Y to be equal to k Thus theprobability that X is in the interval …x; x ‡ dx† is

f …x† dx But this probability is equal to the probability

of there having occurred …k 1† Poisson events in aperiod of duration x times the probability of occur-rence of one event in a period of duration dx Thus,

we have

f …x† dx ˆe …k 1†!x…x†k 1  dxfrom which we obtain

f …x† ˆ…x†…k 1†!k 1e x 0  x < 1 …12†Expression (12), taking into account that the gammafunction for an integer k satis®es

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Table 3 Some Continuous Probability Density Functions that Arise in EngineeringApplications

1 < x < 1Central F ……n1‡ n2†=2†nn1 =2

particular cases of the beta distribution are interesting.

Setting (r=1, s=1), gives the standard uniform U…0; 1†

distribution, while setting …r ˆ 1; s ˆ 2 or r ˆ 2; s ˆ 1)

gives the triangular random variable whose cdf is given

1.5.5 The Normal or Gaussian Distribution

One of the most important distributions in probability

and statistics is the normal distribution (also known as

the Gaussian distribution), which arises in various

applications For example, consider the random

vari-able, X, which is the sum of n identically and

indepen-dently distributed (iid) random variables Xi Then, by

the central limit theorem, X is asymptotically normal,

regardless of the form of the distribution of the

ran-dom variables Xi

The normal random variable with parameters  and

2 is denoted by N…; 2† and its pdf is

f …x† ˆ 1

p2 exp

…x †222

!

1 < x < 1

The change of variable, Z ˆ …X †=, transforms

a normal N…; 2† random variable X in another

ran-dom variable Z, which is N…0:1† This variable is called

the standard normal random variable The main

inter-est of this change of variable is that we can use tables

for the standard normal distribution to calculate

prob-abilities for any other normal distribution For

exam-ple, if X is N…; 2†, then

p…X < x† ˆ p X <x 

ˆ p Z < x  ˆ x  where …z† is the cdf of the standard normal distribu-

tion The cdf …z† cannot be given in closed form

However, it has been computed numerically and tables

for …z† are found at the end of probability and

statis-tics textbooks Thus we can use the tables for the

stan-dard normal distribution to calculate probabilities for

any other normal distribution

1.5.6 The Log-Normal Distribution

We have seen in the previous subsection that the sum

of iid random variables has given rise to a normaldistribution In some cases, however, some randomvariables are de®ned to be the products instead ofsums of iid random variables In these cases, takingthe logarithm of the product yields the log-normal dis-tribution, because the logarithm of a product is thesum of the logarithms of its components Thus, wesay that a random variable X is log-normal when itslogarithm ln X is normal

Using Theorem 7, the pdf of the log-normal randomvariable can be expressed as

f …x† ˆ 1xp2 exp

…ln x †222

!

x  0

where the parameters  and  are the mean and thestandard deviation of the initial random normal vari-able The mean and variance of the log-normal ran-dom variable are e‡ 2 =2 and e2…e2 2

e 2

†,respectively

1.5.7 The Chi-Squared and Related DistributionsLet Y1; ; Yn be independent random variables,where Yi is distributed as N…i; 1† Then, the variable

X ˆXn

iˆ1

Yi2

is called a noncentral chi-squared random variable with

n degrees of freedom, noncenrality parameter

 ˆPniˆ12i; and is denoted as 2n…† When  ˆ 0 weobtain the central chi-squared random variable, which

is denoted by 2n The pdf of the central chi-squaredrandom variable with n degrees of freedom is given in

Table 3, where …:† is the gamma function de®ned in

Eq (13)

The positive square root of a 2

n…† random variable

is called a chi random variable and is denoted by

n…† An interesting particular case of the n…† isthe Rayleigh random variable, which is obtained for

…n ˆ 2 and  ˆ 0† The pdf of the Rayleigh randomvariable is given in Table 3 The Rayleigh distribution

is used, for example, to model wave heights [5].1.5.8 The t Distribution

Let Y1be a normal N…; 1† and Y2be a 2independentrandom variables Then, the random variable

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