Mathematical statistics for economics and business (second edition) part 1

Preface to the Second Edition of Mathematical Statistics for Economics and Business n n The general objectives of the second edition of to provide a rigorous and accessible foundation in

Mathematical Statistics for

Economics and Business

Ron C Mittelhammer

Second Edition

Mathematical Statistics

for Economics and Business Second Edition

Ron C Mittelhammer

Mathematical Statistics

for Economics and Business

Second Edition With 93 Illustrations

Ron C Mittelhammer

School of Economic Sciences

Washington State University

Pullman, Washington

USA

ISBN 978-1-4614-5021-4 ISBN 978-1-4614-5022-1 (eBook)

DOI 10.1007/978-1-4614-5022-1

Springer New York Heidelberg Dordrecht London

Library of Congress Control Number: 2012950028

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and to the memory of Carl, Edith, Dolly, and Ralph.

Preface to the Second Edition

of Mathematical Statistics for Economics and Business

n

n

The general objectives of the second edition of

to provide a rigorous and accessible foundation in the principles of probabilityand in statistical estimation and inference concepts for beginning graduatestudents and advanced undergraduate students studying primarily in the fields

of economics and business Since its publication, the first edition of the book hasfound use by those from other disciplines as well, including the social sciences(e.g., psychology and sociology), applied mathematics, and statistics, eventhough many of the applied examples in later chapters have a decidedly “eco-nomics and business” feel (although the examples are chosen in such a way thatthey are fairly well “self-contained” and understandable for those who have notstudied either discipline in substantial detail)

The general philosophy regarding how and why the book was originallywritten was presented in the preface to the first edition and in large measurecould be inserted at this point for motivating the fundamental rationale for thesecond edition This philosophy includes the necessity of having a conceptualbase of probability and statistical theory to be able to fully understand theapplication and interpretation of applied econometric and business statisticsmethods, coupled with the need to have a treatment of the subject that, whilerigorous, also assumes an accessible level of prerequisites that can be expected

to have been met by a large majority of graduate students entering the fields.The choice of topic coverage is also deliberate and decidedly chosen to formthe fundamental foundation on which econometric and business statisticsmethodology is built With the ongoing expansion, in both scope and depth,

of econometric and statistical methodology for quantitative analyses in botheconomics and business, it has never been more important, and many are nowthinking absolutely essential, that a base of formal probability and statisticsunderstanding become part of student training to enable effective reading of theliterature and success in the fields.

Regarding the nature of the updates and revisions that have been made inproducing the second edition, many of the basic probability and statisticalconcepts remain in common with the first edition The fundamental base ofprobability and statistics principles needed for later study of econometrics,business statistics, and a myriad of stochastic applications of economic andbusiness theory largely intersects the topics covered in the first edition While

a few topics were deleted in the second edition as being less central to thatfoundation, many more have been added These include the following: greaterdetail on the issue of parametric, semiparametric, and nonparametric models; anintroduction to nonlinear least squares methods; Stieltjes integration has beenadded strategically in some contexts where continuous and discrete randomvariable properties could be clearly and efficiently motivated in parallel; addi-tional testing methodology for the ubiquitous normality assumption; clearerdifferentiation of parametric and semiparametric testing of hypotheses; as well

as many other refinements in topic coverage appropriate for applications ineconomics and business

Perhaps the most important revision of the text has been in terms of theorganization, exposition, and overall usability of the material Reacting to thefeedback of a host of professors, instructors, and individual readers of the firstedition, the presentation of both the previous and new material has been notablyreorganized and rewritten to make the text easier to study and teach from At thehighest level, the compartmentalization of topics is now better and easier tonavigate through All theorems and examples are now titled to provide a betterforeshadowing of the content of the results and/or the nature of what is beingillustrated Some topics have been reordered to improve the flow of reading andunderstanding (i.e., the relatively more esoteric concept of events that cannot beassigned probability consistently has been moved to the end of a chapter and thereview of elements of real analysis has been moved from the beginning of theasymptotic theory chapter to the appendix of the book), and in some cases,selected proofs of theorems that were essentially pure mathematics and thatdid little to bolster the understanding of statistical concepts were moved tochapter appendices to improve readability of the chapter text A large number

of new and expanded exercises/problems have been added to the chapters.While a number of texts focused on statistical foundations of estimation andinference are available, Mathematical Statistics for Economics and Business is atext whose level of presentation, assumed prerequisites, examples and problems,and topic coverage will continue to provide a solid foundation for future study ofeconometrics, business statistics, and general stochastic economic and businesstheory and application With its redesigned topic organization, additional topiccoverage, revision of exposition, expanded set of problems, and continued focus

on accessibility and motivation, the book will provide a conceptual foundation

viii Preface to the Second Edition of Mathematical Statistics for Economics and Business

on which students can base their future study and understanding of rigorouseconometric and statistical applications, and it can also serve as an accessiblerefresher for practicing professionals who wish to reestablish their understand-ing of the foundations on which all of econometrics, business statistics, andstochastic economic and business theory are based.

Acknowledgments

In addition to all of the acknowledgments presented in thefirst edition, which certainly remain deserving of inclusion here, I would like tothank Ms Danielle Engelhardt, whose enormous skills in typing, formatting,and proof-checking of the text material and whose always cheerful and positive

“can-do” personality made the revision experience a much more enjoyable andefficient process I am also indebted to Dr Miguel Henry-Osorio for proofreadingevery character of every page of material and pointing out corrections, in addi-tion to making some expositional suggestions that were very helpful to therevision process Mr Sherzod Akhundjanov also provided expert proof-checking,for which I am very grateful I also thank Haylee and Hanna Gecas for theirconstant monitoring of my progress on the book revision and for makingsure that I did not stray too far from the targeted timeline for the effort I alsowish to thank my colleague Dr Tom Marsh, who utilized the first edition of thisbook for many years in the teaching of his econometrics classes and whoprovided me with helpful feedback on student learning from and topic coverage

in the book Finally, a deep thank you for the many comments and helpfulsuggestions I continued to receive over the years from my many doctoralstudents, the students who attended my statistics and econometrics classeshere at the university; the many additional questions and comments I receivedfrom students elsewhere; and the input received from a host of individuals allover the world – the revision of the book has benefitted substantially from yourinput Thank you all

statisti-Without the unifying foundations that come with training in probability andmathematical statistics, students in statistics and econometrics classes toooften perceive the subject matter as a potpourri of formulae and techniquesapplied to a collection of special cases The details of the cases and theirsolutions quickly fade for those who do not understand the reasons for usingthe procedures they attempt to apply Many institutions now recognize the needfor a more rigorous study of probability and mathematical statistics principles inorder to prepare students for a higher-level, longer-lasting understanding of thestatistical techniques employed in the fields of business and economics Fur-thermore, quantitative analysis in these fields has progressed to the point where

a deeper understanding of the principles of probability and statistics is nowvirtually necessary for one to read and contribute successfully to quantitativeresearch in economics and business Contemporary students themselves knowthis and need little convincing from advisors that substantial statistical trainingmust be acquired in order to compete successfully with their peers and tobecome effective researchers Despite these observations, there are very fewrigorous books on probability and mathematical statistics foundations that arealso written with the needs of business and economics students in mind.This book is the culmination of 15 years of teaching graduate level statisticsand econometrics classes for students who are beginning graduate programs inbusiness (primarily finance, marketing, accounting, and decision sciences), eco-nomics, and agricultural economics When I originally took on the teachingassignment in this area, I cycled through a number of very good texts in mathe-matical statistics searching for an appropriate exposition for beginning graduatestudents With the help of my students, I ultimately realized that the availabletextbook presentations were optimizing the wrong objective functions for ourpurposes! Some books were too elementary; other presentations did not covermultivariate topics in sufficient detail, and proofs of important results wereomitted occasionally because they were “obvious” or “clear” or “beyond thescope of the text.” In most cases, they were neither obvious nor clear to students,and in many cases, useful and accessible proofs of the most important resultscan and should be provided at this level of instruction Sufficient asymptotictheory was often lacking and/or tersely developed At the extreme, material waspresented in a sterile mathematical context at a level that was inaccessible tomost beginning graduate students while nonetheless leaving notable gaps intopic coverage of particular interest to business and economics students Notingthese problems, gaps, and excesses, I began to teach the course from lecturenotes that I had created and iteratively refined them as I interacted with scores ofstudents who provided me with feedback regarding what was working—andwhat wasn’t—with regard to topics, proofs, problems, and exposition I amdeeply indebted to the hundreds of students who persevered through, andcontributed to, the many revisions and continual sophistication of my notes.Their influence has had a substantial impact on the text: It is a time-tested andclass-tested product Other students at a similar stage of development shouldfind it honest, accessible, and informative.

Instructors attempting to teach a rigorous course in mathematical statisticssoon learn that the typical new graduate student in economics and business isthoroughly intelligent, but often lacks the sophisticated mathematical trainingthat facilitates understanding and assimilation of the mathematical conceptsinvolved in mathematical statistics My experience has been that these studentscan understand and become functional with sophisticated concepts in mathe-matical statistics if their backgrounds are respected and the material ispresented carefully and thoroughly, using a realistic level of mathematics Fur-thermore, it has been my experience that most students are actually eager to seeproofs of propositions, as opposed to merely accepting statements on faith, solong as the proofs do not insult the integrity of the nonmathematician

xii Preface

Additionally, students almost always remark that the understanding and thelong-term memory of a stated result are enhanced by first having workedthrough a formal proof of a proposition and then working through examplesand problems that require the result to be applied.

With the preceding observations in mind, the prerequisites for the bookinclude only the usual introductory college-level courses in basic calculus(including univariate integration and differentiation, partial differentiation,and multivariate integration of the iterated integral type) and basic matrixalgebra The text is largely self-contained for students with this preparation

A significant effort has been made to present proofs in ways that are accessible.Care has been taken to choose methods and types of proofs that exercise andextend the learning process regarding statistical results and concepts learnedprior to the introduction of the proof A generous number of examples arepresented with a substantial amount of detail to illustrate the application ofmajor theories, concepts, and methods The problems at the end of the chaptersare chosen to provide an additional perspective to the learning process Themajority of the problems are word problems designed to challenge the reader

to become adept at what is generally the most difficult hurdle—translatingdescriptions of statistical problems arising in business and economic settingsinto a form that lends itself to solutions based on mathematical statisticsprinciples I have also warned students through the use of asterisks (*) when aproof, concept, example, or problem may be stretching the bounds of theprerequisites so as not to frustrate the otherwise diligent reader, and to indicatewhen the help of an instructor or additional readings may be useful

The book is designed to be versatile The course that inspired this book is asemester-long four-credit intensive mathematical statistics foundation course

I do not lecture on all of the topics contained in the book in the 50 contact hoursavailable in the semester The topics that I do not cover are taught in the firsthalf of a subsequent semester-long three-credit course in statistics and econo-metric methods I have tended to treat Chapters 1–4 in detail, and I recommendthat this material be thoroughly understood before venturing into the statisticalinference portion of the book Thereafter, the choice of topics is flexible Forexample, the instructor can control the depth at which asymptotic theory istaught by her choice of whether the starred topics in Chapter 5 are discussed.While random sampling, empirical distribution functions, and sample momentsshould be covered in Chapter 6, the instructor has leeway in the degree ofemphasis that she places on other topics in the chapter Point estimation andhypothesis testing topics can then be mixed and matched with a minimalamount of back-referencing between the respective chapters

Distinguishing features of this book include the care with which topics areintroduced, motivated, and built upon one another; use of the appropriate level

of mathematics; the generous level of detail provided in the proofs; and a familiarbusiness and economics context for examples and problems This text is bitlonger than some of the others in the field The additional length comes fromadditional explanation, and detail in examples, problems, and proofs, and notfrom a proliferation of topics which are merely surveyed rather than fully

developed As I see it, a survey of statistical techniques is useful only after onehas the fundamental statistical background to appreciate what is being sur-veyed And this book provides the necessary background.

Acknowledgments

I am indebted to a large number of people for their encouragement andcomments Millard Hastay, now retired from the Washington State Universityeconomics faculty, is largely responsible for my unwavering curiosity andenthusiasm for the field of theoretical and applied statistics and econometrics.George Judge has been a constant source of encouragement for the book projectand over the years has provided me with very valuable and selfless advice andsupport in all endeavors in which our paths have crossed I thank Jim Chalfantfor giving earlier drafts of chapters a trial run at Berkeley, and for providing mewith valuable student and instructor feedback Thomas Severini at Northwest-ern provided important and helpful critiques of content and exposition MartinGilchrist at Springer-Verlag provided productive and pleasurable guidance to thewriting and revision of the text I also acknowledge the steadfast support ofWashington State University in the pursuit of the writing of this book Of themultitude of past students who contributed so much to the final product andthat are too numerous to name explicitly, I owe a special measure of thanks toDon Blayney, now of the Economic Research Service, and Brett Crow, currently

a promising Ph.D candidate in economics at WSU, for reviewing drafts of thetext literally character by character and demanding clarification in a number ofproofs and examples I also wish to thank many past secretaries who toiledfaithfully on the book project In particular, I wish to thank Brenda Campbell,who at times literally typed morning, noon, and night to bring the manuscript tocompletion, without whom completing the project would have been infinitelymore difficult Finally, I thank my wife Linda, who proofread many parts of thetext, provided unwavering support, sustenance, and encouragement to methroughout the project, and despite all of the trials and tribulations, remains

my best friend

xiv Preface

2.2.3 Continuous Random Variables and Probability Density

2.8 Extended Example of Multivariate Concepts

xvi Contents

3.5 Conditional Expectation 125

3.11 Means and Variances of Linear Combinations

3.12 Appendix: Proofs and Conditions for Positive

4.2.3 Gamma Subfamily Name: Exponential 195

5.4 Convergence in Mean Square (or Convergence

5.5 Almost-Sure Convergence (or Convergence

5.6 Summary of General Relationships Between

xviii Contents

5.9 Asymptotic Distributions of Differentiable Functions

of Asymptotically Normally Distributed Random Variables:

7.1 Parametric, Semiparametric, and Nonparametric

7.1.3 Nonparametric Models 366

7.2 Additional Considerations for the Specification

xx Contents

8.3.3 MLE Properties: Large Sample 479

8.4 Method of Moments and Generalized Method of Moments

9.5 Classical Hypothesis Testing Theory: UMP

10 Hypothesis Testing Methods and Confidence Regions 609

Math Review Appendix: Sets, Functions,

xxii Contents

Relation (Binary) 708

1 1ffiffiffiffi2p

the number of degrees of freedom (v) The headings

of the other columns give probabilities (P) for t

to exceed the entry value Use symmetry

the number of degrees of freedom (v) The headings

v

The headings of the other columns list the numerator

The headings of the other columns list the numerator

List of Figures

n

n

Figure 4.5 Gamma densities, case I, b ¼ 1 193

Figure 9.9 Power function for the UMPU level 057 test

List of Tables

n

n

n

Elements of Probability Theory

n n

n

that a shipment of smart phones contains less than 5 percent defectives, that agambler will win a crap game, that next year’s corn yields will exceed 80 bushelsper acre, or that electricity demand in Los Angeles will exceed generatingcapacity on a given day This probability concept will also be relevant forquantifying the degree of belief in such propositions as it will rain in Seattletomorrow, Congress will raise taxes next year, and the United States will sufferanother recession in the coming year

The value of such a measure of outcome propensity or degree of belief can besubstantial in the context of decision making in business, economics, govern-ment, and everyday life In the absence of such a measure, all one can effectivelysay when faced with an uncertain situation is “I don’t know what will happen”

or “I don’t know whether the proposition is true or false.” A rational maker will prefer to have as much information as possible about the finaloutcome or state of affairs associated with an uncertain situation in order tomore fully consider its impacts on profits, utility, welfare, or other measures ofwell-being Indeed, the problem of increasing profit, utility, or welfare through

decision-appropriate choices of production and inventory levels and scheduling, productpricing, advertising effort, trade policy, tax strategy, input or commoditypurchases, technology adoption, and/or capital investment is substantiallymore difficult when the results of one’s choice are affected by factors that aresimply unknown, as opposed to occurring with varying degrees of likelihood.Probability is a tool for distinguishing likely from unlikely outcomes orstates of affairs and provides business managers, economists, legislators,consumers, and individuals with information that can be used to rank thepotential results of their decisions in terms of propensity to occur or degree ofvalidity It then may be possible to make choices that maximize the likelihood of

a desired outcome, provide a high likelihood of avoiding disastrous outcomes, orachieve a desirable expected result (where “expected” will be rigorously defined

in the next chapter)

There have been many ways proposed for defining the type of quantitativemeasure described above, and that are useful in many respects, but they have not

functions (Schaefer), structural probability (Fraser), and fiducial probability(Fisher) Four principal definitions that have found substantial degrees of accep-tance and use, and that have been involved in the modern development of

subjective probability, and the axiomatic approach We will briefly discuss thefirst three definitions, and then concentrate on the modern axiomatic approach,which will be seen to subsume the other three approaches as special cases.Prior to our excursion into the realm of probability theory, it is helpful toexamine how the terms “experiment,” “sample space,” “outcome,” and “event”will be used in our discussion The next section provides the necessaryinformation

statistics, and is not at all limited to the colloquial interpretation of the term

as referring to activities that scientists perform in laboratories

Definition 1.1

states of affairs can be identified

Thus, determining the yield per acre of a new type of wheat, observing thequantity of a commodity sold during a promotional campaign, identifying the fatpercentage of a hundredweight of raw farm milk, observing tomorrow’s closingDow Jones Industrial Average on the NY Stock Exchange, or analyzing theunderlying income elasticity affecting the demand for gasoline are all examples

of experiments according to this definition of the term

The final state of affairs resulting from an experiment is referred to as anoutcome.

Definition 1.2

Outcome of an

Experiment

A final result, observation, or measurement occurring from an experiment

Thus, referring to the preceding examples of experiments, 80 bushels peracre, 2,500 units sold during a week of promotions, 3.7 percent fat per hundred-weight, a DJIA of 13,500, and an income elasticity of 75 are, respectively,possible outcomes

Prior to analyzing probabilities of outcomes of an experiment, it is necessary

space of an experiment

Definition 1.3

Note that our definition of sample space, which we will henceforth denote

sample space contains all possible outcomes of an experiment In many cases,the set of all possible outcomes will be readily identifiable and not subject tocontroversy, and in these cases it will be natural to refer to this set as the samplespace For example, the experiment of rolling a die and observing the number ofdots facing up has a sample space that can be rather uncontroversially specified

as {1, 2, 3, 4, 5, 6} (as long as one is ruling out that the die will not land on anedge!) However, defining the collection of possible outcomes of an experimentmay also require some careful deliberation For instance, in our example ofmeasuring the fat percentage of a given hundredweight of raw farm milk, it is

the accuracy of our measuring device might only allow us to observe differences

in fat percentages up to hundredths of a percent, and thus a smaller set

0; :01; :02; :::; 100

of greater than 20 percent and less than 1 percent will simply not occur in raw

could represent the sample space of the fat-measuring experiment Fortunately,

as the reader will come to recognize, the principal concern of practical tance is that the sample space be specified large enough to contain the set of allpossible outcomes of the experiment as a subset The sample space need not beidentically equal to the set of all possible outcomes The reader may wish tosuggest appropriate sample spaces for the remaining four example experimentsdescribed above

impor-Consistent with set theory terminology and the fact that the sample space isindeed a set, each outcome in a sample space can also be referred to as anelement or member of the sample space In addition, the outcomes in a sample

space are also sometimes referred to as sample points The reader should beaware of these multiple names for the same concept, and there will be otherinstances ahead where concepts are referred to by multiple different names.The sample space, as all sets, can be classified according to whether the

study due to the fact that probabilities will ultimately be assigned using eitherfinite mathematics, or via calculus, respectively

An uncountably infinite sample space that consists of a continuum of points

The fundamental entities to which probabilities will be assigned are events,which are equivalent to subsets in set theoretic terminology

Definition 1.6

Thus, events are simply collections of outcomes of an experiment Note that

a technical issue in measure theory can arise when we are dealing withuncountably infinite sample spaces, such that certain complicated subsets can-not be assigned probability in a consistent manner For this reason, in more

refer to measureable subsets of the sample space We provide some backgroundrelating to this theoretical problem in the Appendix of this chapter As a practi-cal matter, all of the subsets to which an empirical analyst would be interested

in assigning probability will be measureable, and we refrain from explicitlyusing this qualification henceforth

In the special case where the event consists of a single element or outcome,

Definition 1.7

1 A countably infinite set is one that has an infinite number of elements that can be “counted” in the sense of being able to place the elements in a one-to-one correspondence with the positive integers An uncountable infinite set has an infinite number of elements that cannot be counted, i.e., the elements of the set cannot be placed in a one-to-one correspondence with the positive integers.

One says that an event A has occurred if the experiment results in anoutcome that is a member or element of the event or subset A.

provided by the real-world definition of the set A That is, verbal or cal statements that are utilized in a verbal or mathematical definition of set A, orthe collection of elements or description of elements placed in brackets in anexhaustive listing of set A, provide the meaning of “the event A has occurred.”

of an event

Example 1.1

Occurrence of

Dice Events

An experiment consists of rolling a die and observing the number of dots facing

whose occurrence means that after a roll, the number of dots facing up on the die

(the actual percentage of the consumer panel preferring Schpitz beer) is such that

the consumers preferred Schpitz to Nickelob or, in other words, the group of

When two events have no outcomes in common, they are referred to asdisjoint events

Definition 1.9

The concept is identical to the concept of mutually exclusive or disjoint sets,

Events that are not mutually exclusive can occur simultaneously Events A1and

We should emphasize that in applications it is the researcher who specifiesthe events in the sample space whose occurrence or lack thereof provides usefulinformation from the researcher’s viewpoint Thus, referring to Example 1.2,

if the researcher were employed by Schpitz Brewery, the identification of whichbeer was preferred by a majority of the participants in a taste comparison wouldappear to be of significant interest to the management of the brewery, and thus

considered important if the advertising department of Schpitz Brewery wished

to utilize an advertising slogan such as “Schpitz beer is preferred to Nickelob bymore than 3 to 1.”

There are three prominent nonaxiomatic definitions of probability that havebeen suggested in the course of the development of probability theory We brieflydiscuss each of these alternative probability definitions In the definition below,N(A) is the size-of-set function whose value equals the number of elements thatare contained in the set A (see Definition A.21)

Definition 1.10

N(A)/N(S)

In the classical definition, probabilities are images of sets generated by a set

subsets of a finite (and thus discrete) sample space, while the range is contained

illustrates the application of the classical probability concept

Example 1.3

Fair Dice and

Classical Probability

Reexamine the die-rolling experiment of Example 1.1, and now assume that the

Therefore, the probability of rolling a three or less and the probability of

experiment is an element of the sample space is 1, as it intuitively should be if

The classical definition has two major limitations that preclude its use asthe foundation on which to build a general theory of probability First, the

for defining the probabilities of events contained in a countably infinite oruncountably infinite sample space Another limitation of the classical definition

is that outcomes of an experiment must be equally likely Thus, for example,

if in a coin-tossing experiment it cannot be assumed that the coin is fair, theclassical probability definition provides no information about how probabilitiesshould be defined In order to relax these restrictions, we examine the relativefrequency approach

Definition 1.11

Relative Frequency

Probability

Let n be the number of times that an experiment is repeated under identical

number of times in n repetitions of the experiment that the event A occurs

It is recognized that in the relative frequency definition, the probability of an

defined as the limiting fraction of the total number of outcomes of the nexperiments that are observed to be members of the set A As in the classical

collection of all subsets of the sample space, while the range is contained in

illustrates the application of the relative frequency concept of probability.Example 1.4

Coin Tossing and

No of tosses No of heads Relative frequency

finite Also, there is no need to assume that outcomes are equally likely, since the

Unfortunately, there are problems with the relative frequency definitionthat reduce its appeal as a foundation for the development of a general theory

of probability First of all, while it is an empirical fact in many types ofexperiments, such as the coin-tossing experiment in Example 1.4, that the

will actually converge to a limit in all cases? Indeed, how could we ever observethe limiting value if an infinite number of repetitions of the experimentare required? Furthermore, even if there is convergence to a limiting value inone sequence of experiments, how do we know that convergence to the samevalue will occur in another sequence of the experiments? Lacking a definitiveanswer to these conceptual queries, we refrain from using the relative frequencydefinition as the foundation for the probability concept

A third approach to defining probability involves personal opinion,

Definition 1.12

the number 1 being associated with certainty

Like the preceding definitions of probability, subjective probabilities can be

obviously vary depending on who is assigning the probabilities and the personalbeliefs of the individual assigning the probabilities Even supposing that twoindividuals possess exactly the same information regarding the characteristics of

an experiment, the way in which each individual interprets the information mayresult in differing probability assignments to an event A

Unlike the relative frequency approach, subjective probabilities can

be defined for experiments that cannot be repeated For example, one might beassigning probability to the proposition that a recession will occur in the comingyear Defining the probability of the event “recession next year” does notconveniently fit into the relative frequency definition of probability, since onecan only run the experiment of observing whether a recession occurs next yearonce In addition, the classical definition would not apply unless a recession, ornot, were equally likely, a priori Similarly, assigning probability to the eventthat one or the other team will win in a Superbowl game is commonly done invarious ways by many individuals, and a considerable amount of betting is based

on those probability assignments However, the particular Superbowl ment” cannot be repeated, nor is there usually any a priori reason to suspect thatthe outcomes are equally likely so that neither relative frequency nor classicalprobability definitions apply

“experi-In certain problem contexts the assignment of probabilities solely on thebasis of personal beliefs may be undesirable For example, if an individual isbetting on some game of chance, that individual would prefer to know the

“true” likelihood of the game’s various outcomes and not rely merely on his orher personal perceptions For example, after inspecting a penny, suppose youconsider the coin to be fair and (subjectively) assign a probability of ½ to each ofthe outcomes “heads” and “tails.” However, if the penny was supplied by aruthless gambler who altered the penny in such a way that an outcome of heads

is twice as likely to occur as tails, the gambler could induce you to bet in such away that you would lose money in the long run if you adhered to your initialsubjective probability assignments and bet as if both outcomes were equallylikely – the game would not be “fair.”

There is another issue relating to the concept of subjective probability thatneeds to be considered in assessing its applicability Why should one assumethat the numbers assigned by any given individual behave in a manner thatmakes sense as a measure of the propensity for an uncertain outcome to occur,

or the degree of belief that a proposition or conjecture is true? Indeed, they mightnot, and we seek criteria that individuals must follow so that the numbers theyassign do make sense as probabilities

Given that objective (classical and relative frequency approaches) and jective probability concepts might both be useful, depending on the problemsituation, we seek a probability theory that is general enough to accommodateall of the concepts of probability discussed heretofore Such an accommodationcan be achieved by defining probability in axiomatic terms

Our objective is to devise a quantitative measure of the propensity of events tooccur, or the degree of belief in various events contained in a sample space Howshould one go about defining such a measure? A useful approach is to define themeasure in terms of properties that are believed to be generally appropriate and/

or necessary for the measure to make sense for its intended purpose So long asthe properties are not contradictory, the properties can then be viewed collec-tively as a set of axioms on which to build the concept of probability.

Note, as an aside, that the approach of using a set of axioms as the tion for a body of theory should be particularly familiar to students of businessand economics For example, the neoclassical theory of the consumer is founded

founda-on a set of behavioral assumptifounda-ons, i.e., a set of axioms The reader might recallthat the axioms of comparability, transitivity, and continuity of preferences aresufficient for the existence of a utility function, the maximization of which,subject to an income constraint, depicts consumption behavior in the neoclassi-

What mathematical properties should a measure of probability possess? First

of all, it seems useful for the measure to be in the form of a real-valued setfunction, since this would allow probabilities of events to be stated in terms ofreal numbers, and moreover, it is consistent with all of the prior definitions ofprobability reviewed in the previous section Thus, we begin with a set function,say P, which has as its domain all of the events in a sample space, S, and has as its

Definition 1.13

We have in mind that the images of events under P will be probabilities of

properties seem appropriate to impose on the real-valued set function P?

recognized that in each case, probability was defined to be a nonnegative ber Since each of the previous nonaxiomatic definitions of probability possesssome intuitive appeal as measures of the propensity of an event to occur or thedegree of belief in an event (despite our recognition of some conceptualdifficulties), let us agree that the measure should be nonnegative valued Bydoing so, we will have defined the first axiom to which the measure must adherewhile remaining consistent with all of our previous probability definitions.Since we decided that our measure would be generated by a set function, P, ourassumption requires that the set function be such that the image of any event A,P(A), be a nonnegative number Our first axiom is thus

num-3 See G Debreu (1959), Theory of Value: An Axiomatic Analysis of Economic Equilibrium Cowles Monograph 17 New York: John Wiley, pp 60–63 Note that additional axioms are generally included that are not needed for the existence of a utility function per se, but that lead to a simplification of the consumer maximization problem See L Phlips (1983), Applied Consumption Analysis New York: North Holland, pp 8–11.