Introduction to Probability Theory and Statistical Inference [Third ed.]

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Probability and Mathematical Statistics ADLER • The Geometry of Random Fields ANDERSON • The Statistical Analysis of Time Series ANDERSON • An Introduction to Multivariate Statistical Analysis ARAUJO and GINE • The Central Limit Theorem for Real and Banach Valued Random Variables ARNOLD • The Theory of Linear Models and Multivariate Analysis BARLOW, BARTHOLOMEW, BREMNER, and BRUNK • Statistical Inference Under Order Restrictions BARNETT • Comparative Statistical Inference BH ATTACH ARYYA and JOHNSON • Statistical Concepts and Methods BILLINGSLEY • Probability and Measure CASSEL, SARNDAL, and WRETMAN • Foundations of Inference in Survey Sampling COCHRAN • Contributions to Statistics DE FINETTI • Theory of Probability, Volumes I and II DOOB • Stochastic Processes FELLER • An Introduction to Probability Theory and Its Applications, Volume I, Third Edition, Revised; Volume II, Second Edition FULLER • Introduction to Statistical Time Series GRENANDER • Abstract Inference HANNAN • Multiple Time Series HANSEN, HURWITZ, and MADOW • Sample Survey Methods and Theory, Volumes I and II HARDING and KENDALL • Stochastic Geometry HOEL • Introduction to Mathematical Statistics, Fourth Edition HUBER • Robust Statistics IOSIFESCU • Finite Markov Processes and Applications ISAACSON and MADSEN • Markov Chains KAGAN, L1NNIK, and RAO • Characterization Problems in Mathematical Statistics KENDALL and HARDING • Stochastic Analysis LAHA and ROHATGI • Probability Theory LARSON • Introduction to Probability Theory and Statistical Inference, Third Edition LARSON • Introduction to the Theory of Statistics LEHMANN • Testing Statistical Hypotheses MATTHES, KERSTAN, and MECKE • Infinitely Divisible Point Processes PARZEN • Modern Probability Theory and Its Applications PURI and SEN • Nonparametric Methods in Multivariate Analysis RANDLES and WOLFE • Introduction to the Theory of Nonparametric Statistics RAO • Linear Statistical Inference and Its Applications, Second Edition ROHATGI • An Introduction to Probability Theory and Mathematical Statistics RUBINSTEIN • Simulation and The Monte Carlo Method SCHEFFE • The Analysis of Variance SEBER • Linear Regression Analysis SEN • Sequential Nonparametrics: Invariance Principles and Statistical Inference SERFLING • Approximation Theorems of Mathematical Statistics TJUR • Probability Based on Radon Measures

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Introduction to Probability Theory and Statistical Inference

Introduction to Probability Theory And Statistical Inference THIRD EDITION

Harold J. Larson Naval Postgraduate School Monterey, California

JOHN WILEY & SONS New York » Chichester • Brisbane • Toronto • Singapore

Copyright

©

1969; 1974, 1982, by John Wiley & Sons, Inc.

All rights reserved. Published simultaneously in Canada. Reproduction or translation of any part of this work beyond that permitted by Sections 107 and 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department, John Wiley & Sons.

Library of Congress Cataloging in Publication Data Larson, Harold J., 1934Introduction to probability theory and statistical inference. (Wiley series in probability and mathematical statistics. Probability and statistics section ISBN 0271-6232). Bibliography: p. Includes index. 1. Probabilities. I. Title. II. Series. QA273.L352

2. Mathematical statistics. 1982

519.2

ISBN 0-471-05909-9 Printed in the United States of America 20 19 18 17 16 15 14 13 12 11

81-16246 AACR2

For Marie

'

Preface This text provides an introduction to probability theory and to many methods used in problems of statistical inference. Some stress is laid on the concept of a probabilistic model for the mechanism generating a set of observed data, leading to the natural application of probability theory to answer questions of interest. As with the previous editions, a one-year course in calculus should allow the student easy access to the material presented. I believe that this method of presentation is ideal for students in the physical and social sciences, as well as in engineering; the examples and exercises are not restricted to any discipline. This edition contains new topics, including the bivariate normal probability law and the /^distribution, and discussions of descriptive statistics (Chapter 6), Cramer-Rao bounds for variances of estimators (Chapter 7), twosample inference procedures (Chapters 7 and 8), the analysis of variance (Chapter 9), and nonparametric procedures (Chapter 10). Most of the material from Chapter 3 on has been totally rewritten and includes new examples and exercises. The first five chapters are again devoted to probability theory, the last chapters devoted to problems of statistical inference. The probability topics, and order of coverage, are much the same as in the earlier editions. Chapter 1 again discusses useful concepts from set theory, whereas Chapters 2 and 3 introduce probability, random variables (both discrete and continuous), and some descriptions of the probability law for a random variable. Chapter 4 presents physical arguments leading to the most commonly encountered (one¬ dimensional) probability laws, including the negative binomial, gamma, and beta distributions. Chapter 5, which discusses jointly distributed random variables, has been tightened up in some senses and is more directly aimed toward results that are important (and frequently used) in problems of statistical inference. This chapter ends with a discussion of some of the probability laws associated with random samples of a normal random variable (;^2,7\ and /'distributions). Chapter 6 is completely new, starting with a section on descriptive statistics. Among other things, this provides a vehicle for discussing subscripted variables and manipulations of such variables, subjects that are (incorrectly) frequently assumed transparent to students. The rest of Chapter 6 describes VII

the concept of a random sample and its use in making inferences about an underlying probability law. Chapter 7 discusses point and interval estimation (including two-sample procedures), and Chapter 8 introduces the Neyman-Pearson theory of tests (including two-sample situations); some discussion is also devoted to the test of significance approach because, in practice, this is a commonly adopted stance. Chapter 9 considers least squares estimation and inference and the analysis of variance for the one- and two-way models. Chapter 10presents an introduction to nonparametric procedures, and Chapter 11 discusses Bayesian approaches; these last three chapters are independent of one another and can be mixed or matched as desired. With the current widespread availability of hand-held calculators (and more powerful equipment), I feel that binomial and Poisson probability calculations are reasonably within the capability of today’s students; tables of these two probability laws have been dropped for this edition. The appendix provides a table of the standard normal distribution function, as well as selected quantiles of certain ^2, T, and F distributions. Answers to all exer¬ cises (except those of a “show-that” or “prove-that” nature) are also presented in the appendix; a solutions manual for all exercises is available from Wiley. The material in this edition is ample for a one-year course in probability theory and statistical inference. Chapters 1 to 5 give a useful one-semester introduction to probability theory (aimed toward statistical inference, not stochastic processes); for a one-quarter introduction to probability, Sections 2.5, 3.4, 3.5, the negative binomial, gamma, and betadistributions(Chapter4), and Section 5.7 can be omitted in part or in whole. Fora two-quarter course on probability and inference, the probability material previously listed (except Section 5.7) and the two-sample procedures in Chapters 7 and 8 could be dropped, and the individual instructor can then choose as he or she wishes (or as time dictates) from Chapters 9, 10, and 11. Iam grateful to all those who used the earlier two editions of this book, especially those who had comments to share and look forward to receiving comments on this third edition. None of the material treated is original with the author; hopefully, the order and method of treatment will help to minimize difficulties in its use in practical problems. My thanks to Bob Lande for typing the manuscript. Harold J. Larson

Contents 1 Set Theory

1

1.1 1.2 1.3 1.4

Set Notation, Equality and Subsets Set Operations Functions Summary

1 6 12 18

2

Probability

19

2.1 2.2 2.3 2.4 2.5

Sample Space; Events Probability Axioms Finite Sample Spaces Counting Techniques Some Particular Probability Problems

21 28 33 39 51

2.6 2.7 2.8

Conditional Probability Independent Events Discrete and Continuous Sample Spaces

59 70 78

2.9

Summary

90

3

Random Variables and Distribution Functions

92

3.1 3.2 3.3 3.4 3.5

Random Variables Distribution Functions and Density Functions Expected Values and Summary Measures Moments and Generating Functions Functions of a Random Variable

92 100 116 129 139

3.6

Summary

193

4 Some Standard probability laws

155

4.1

155

The Bernoulli and Binomial Probability Laws IX

4.2 4.3 4.4

Geometric and Negative Binomial Probability Laws Sampling and the Hypergeometric Probability Law The Poisson Probability Law

166 175 183

4.5 4.6 4.7

The Uniform, Exponential and Gamma Probability Laws The Beta and Normal Probability Laws Summary

192 203 220

5

Jointly Distributed Random Variables

222

5.1 5.2 5.3 5.4

Vector Random Variables Conditional Distributions and Independence Expected Values and Moments Sums of Random Variables

223 236 250 264

5.5 5.6

277

5.7 5.8

The Chebyshev Inequality and the Law of Large Numbers The Central Limit Theorem and Probability Law Approximations Some Special Distributions Summary

286 303 323

6

Descriptive and Inferential Statistics

327

6.1 6.2 6.3

Descriptive Statistics Inferential Statistics Summary

327 342 358

7

Estimation of Parameters

359

7.1 7.2 7.3 7.4 7.5

Estimation by Method of Moments and by Maximum Likelihood Properties of Estimators Confidence Intervals Two-Sample Procedures Summary

382 398 409

8

Tests of Hypotheses

411

8.1

360

371

413

Simple Hypotheses

x

8.2

Composite Hypotheses

429

8.3 8.4

Some Two-Sample Tests Summary

444 461

9

Least Squares and Regression

403

9.1

Least Squares Estimation

403

9.2

Interval Estimation and Tests of Hypotheses The Analysis of Variance Summary

4g2

9.3 9.4

I 0 10.1 10.2 10.3 10.4 10.5/

Nonparametric Methods Inferences about Quantiles The Run Test and the Wilcoxon-ManmWhitney Test Chi-Square Goodness of Fit Tests Contingency Tables Summary

Bayesian Methods

II

III 11.2 11.3 11.4

481 0Qg

508 509 518 527 541 551

552

Prior and Posterior Distributions Bayesian Estimators Bayesian Intervals Summary

Appendix

553 561 569 577

579

Table 1

Standard Normal Distribution Function

580

Table 2 Table 3

Chi-Square Quantiles T Distribution Quantiles

582 584

Table 4

F Distribution Quantiles

585

Answers to Exercises index

591 629 XI

Introduction to Probability Theory and Statistical Inference

1 Set Theory

In the study of probability theory and statistics an exact medium of communication is extremely important; if the meaning of the question that is asked is confused by semantics, the solution is all the more difficult, if not impossible, to find. The usual exact language employed to state and solve probability problems is that of set theory. The amount of set theory that is required for relative ease and comfort in probability manipulations is easily acquired. We will look briefly at some of the simpler definitions, operations, and concepts of set theory, not because these ideas are neces¬ sarily a part of probability theory but because the time needed to master them is more than compensated for by later simplifications in the study of probability.

1.1

Set Notation, Equality, and Subsets

A set is a collection of objects. The objects themselves can be anything from numbers to battleships. An object that belongs to a particular set is called an element of that set. We will commonly use uppercase letters from the beginning of the alphabet to denote sets )A, B, C, etc.) and lowercase letters from the end of the alphabet to denote elements of sets (x, y, z, etc.). To specify that certain objects belong to a given set, we will use braces { } (commonly called set builders) and either the roster (complete listing of 1

2

Set Theory

all elements) or the rule method. For example, if we want to write that the set A consists of the letters a, b, c and that the set B consists of the first 10 integers, we may write A = {a, b, c} (roster method of specification) B = {x: x = 1, 2, 3,..., 10} (rule method of specification). These two sets can easily be read as “A is the set of elements a, b, c” and “B is the set of elements x such that x = 1 or x = 2 or x = 3 and so on up to x = 10.” We will use the symbol e as shorthand for “belongs to” and thus can write for the two sets defined here that a e A, 7 e B Just as a line drawn through an equals sign is taken as negation of the equality, we will use ^ to mean “does not belong to”; thus a

, |r| 1.

This density function is graphed in Figure 3.2.8. Since the density function for X is equal to 1 for 0 < t < 1, the area under the density over any interval between 0 and 1 is equal to the length of the interval; thus the probability that the observed value for X lies in an interval contained in (0, 1) is proportional to the length of the interval.

12

Random Variables and Distribution Functions

0

b

a

Figure 3.2.8

This is the same method for computing probabilities for continuous sample spaces as was discussed in Section 2.8. Because the density function is constant or uniform over (0, 1), the random variable X is called uniform; we will discuss uniform random variables in Chapter 4. We will somewhat inaccurately say the values from the range of a uniform random variable are “equally likely” to occur. It provides a continuous analog to the discrete equally likely case already discussed. Now let Y be the continuous random variable whose distribution function is H2(t) defined in Example 3.2.5. Fy(t) = H2(t) = 0, = I - e~x.

t < 0 t > 0.

The density function for Y is

which is graphed in Figure 3.2.9. As we will see in Chapter 4. Y is a particu¬ lar example of an exponential random variable. | Let us conclude this section by commenting that any function h(t) can be considered the probability density function of some continuous random variable if

h(t) > 0.

for all f.

and

h(t)dt = 1. — oo

3.2

Distribution Functions and Density Functions

113

fy(t)

EXERCISE 3.2 1.

Verify that

Fx(t) = 0,

to

is a distribution function and derive the probability function for X. Verify that

Fz(t) = 0,

t < —2 -2 < t < 0 t > 0

= i =1,

is a distribution function and specify the probability function for Z. Use it to compute P(— 1 < Z < 1). 3. Verify that

Fw(t) = 0. _ i ~

3

’

_ I —

_ —

2' 2 3’

t < 3 3 < t < 4 4 < t < 5 5 < t < 6 t > 6

is a distribution function and specify the probability function for W. Use it to compute P{3 < W < 5).

114

Random Variables and Distribution Functions

4. Verify that

t < 0 0 < t < 5 < t < 7 < t < 100 < t t > 102

FY{t) = 0,

5.

is a distribution function and specify the probability function for Y. Use it to compute P(Y < 100). The random variable Z has the probability function

pz(x) =

for x = 0, 1, 2 otherwise.

= 0,

6.

5 7 100 < 102

What is the distribution function for Z? The random variable U has the probability function Pu(~ 3) = i Pu(0) = i Pi/(4) =

b

7.

What is the distribution function for U1 Verify that Fx(t)= 0,

=1.

8.

t > 1

is a distribution function and specify the probability density function for X. Use it to compute P( — j < X < j). Verify that Fy(t)

=0.

t < 0

= y/t,

0 < t < 1 t > 1

=1.

9.

t 1

= f,

= 1,

is a distribution function. This is the distribution function of what is called a mixture random variable. It has one point of discontinuity (at t = 0) and thus there is a positive probability (j) of the random variable equaling zero. Note that the distribution function is continu¬ ous and increasing for j < t < 1. Thus the random variable takes on particular values in this interval with probability zero; we could de¬ fine a pseudo density function for the random variable lying in sub¬ intervals of this interval. 12. Given X has probability density function

fx(x) =1, = 0,

99 < x < 100 otherwise,

derive Fx(t). 13. Y is a continuous random variable with

JY(y) = 2(1 — y), = 0,

14.

Derive Fy(t). Z is a continuous random variable with probability density function /z(z) = 10c“ 10:, = 0,

15. 16.

0 < y < 1 otherwise.

z > 0 otherwise.

Derive Fz(t). Let X be the random variable whose density function is defined in Exercise 2.8.12 and derive Fx{t). Let Y be the random variable whose density function is defined in Exercise 2.8.13 and derive FY(t)■

116

Random Variables and Distribution Functions

3.3

Expected Values and Summary Measures

The probability law for a random variable can be defined by its distribution function, Fx(t), or its density function, fx(t), if X is continuous, or its probability function, px(k), if X is discrete. Once we know the prob¬ ability law for A", we are able to compute the probabilities of occurrence for any events of interest. In many applications we will be interested in describing various aspects of different probability laws, ways of describing certain properties of probability laws. For example, what is a “typical” value for the random variable to equal, where “typical” may be defined in various ways. How much variability is exhibited by the probability law, how spread out are the possible observed values for the random variable? In this section we will study some common measures of certain aspects of probability laws, concentrating on measures of the “middle” of the probability law (the typical observed value) and of the “variability” of the probability law (the spread of the possible observed values). Let us discuss the expected or average value of a random variable. A specific gambling example is perhaps the easiest case to visualize first. The game of Chuck-a-Luck was discussed in Example 3.1.6. Briefly, $1 is bet and the amount won is a discrete random variable V with probability function Pv(v) = ill

=

216

= YU = JT6

at v = -1 at V = 1 at v = 2 at v = 3.

Suppose you were to play this game n > 1 times, winning vt dollars the first time, v2 dollars the second time. ..., vn dollars the nth time (remember each Vi can be any of the values 1, 1, 2, 3). The average amount won over these n plays then is —

v

Vf

What might we expect for the value of f ? Since each vt must equal — 1, 1, 2, or 3, let be the number of times in the n plays that you win -1 dollars (you lose your dollar), let k2 be the number of times you win 1 dollar, k3 the number of times you win 2 dollars, and /c4 the number of times you win 3 dollars. Then your average winnings over the n plays is Z vi = (- !) = i

ki n

+ (!)— + (2) — + (3) —.

n

n

n

3.3

Expected Values and Summary Measures

117

The ratio kx/n is the relative frequency in your n plays that you win — 1 dollars; but this number should be roughly equal to jj| = pK(—1) = P{V = — 1), because probabilities should equal relative frequencies of occurrence. Similarly, we would expect that

n

= PvO)

and thus we should find the average amount won to approximately be given by (-l)M-l) + (1)Pk(1) + (2) pv(2) + (3) pv(3) = (- l)(ifi) + UMrre) + (2)(^) + (3) (yf^)

You should expect, on the average, to lose about 8 0 with density function fT(t) pictured in Figure 3.3.1,

/V- ax because Y has “more room” to vary. This does not neces¬ sarily prove true, since the distribution of probability has a great effect on c\\ the random variable with more probability close to has the smaller variance. The following example should help make the point. 3.3.2. Consider random variables U, V, and W with prob¬ ability laws defined by the density functions example

fv(u) = u -1-1, = 1 — u, = 0,

= 0, Jw(w)

- 1 < w < 0 0 < u < 1 otherwise, -1 < v < 1 otherwise.

= w.

-1 < w < 0 0 < w < 1

= 0,

otherwise.

= -w.

3.3

Expected Values and Summary Measures

1 23

These are graphed in Figure 3.3.3. Each density is positive over the interval — 1 to 1 so the observed values for all three random variables can vary over equal length intervals and all three mean values are zero. It is obvious from the picture that we should have al < crj> < even though the ranges are equal. You can verify that in fact „2

_ I

°U ~ 6'

2

_ I

aV ~ 3'

2 1. °V — 2 >

the standard deviations are 1 /,/6, 1 /v/3, and 1 /^2, respectively. ■ An alternative formula for computing the variance of a random variable is a\ = E[(X - /rx)2] = £[Y2 -

+ /.2]

= £[XJ]-2^E[X] + £[,ii]

= £[*2] - Aso the variance can also be thought of as the average of the squares of the values of the random variable less the square of the average. This alternative formula is an easier way to evaluate a\ in almost all cases. Linear transformations of random variables occur quite frequently in applications. Suppose T is the (random) time needed in minutes to com¬ plete a task and Thas mean nT and variance o\; the time needed to complete

Random Variables and Distribution Functions

the task in hours then is H — T/60. Intuitively, it would seem that the mean of H and variance of H should be rather simply related to nT and oF. Or if F is the maximum temperature (in degrees Fahrenheit) to be recorded in Washington, D.C., tomorrow, then the maximum temperature in degrees Celsius is C = f F — again a linear relation. Granted nF and oF are known, can we also evaluate /rc and acl Theorem 3.3.2, which you are asked to prove in Exercise 3.3.17, shows what these relationships are.

Theorem 3.3.2. Assume A is a random variable with mean nx and variance o\. If Y = aX + b, where a and b are any constants, then HY = aqx + b,

o\ = a2ax,

oY = |a| ox.

g

The quantiles of a continuous distribution offer alternative (and per¬ haps more transparent) ways of summarizing or describing probability laws. These are now defined. 3.3.4. Let A be a continuous random variable with distribu¬ tion function Fx(t). The (\00k)th quantile for X is the number tk such that Definition

Fx(tk) = k,

■

0 < k < 1.

The (100A:) th quantile is that value tk that makes the value of the distribution function equal to k; alternatively, it is the value such that the area under the density, to its left, is equal to k. Figure 3.3.4 illustrates both these statements for the random variable U whose density is defined in Example 3.3.2. The 50th quantile, 15, is also called the median for the random variable (really its probability law). It is the value that cuts the density function into two pieces with equal area, .5 to the left and .5 to the right. The median is an alternative measure of the middle of a probability law. The 25th, 50th,

F^t)

fu'u)

o

Figure 3.3.4

lk

1

U

3.3

Expected Values and Summary Measures

1 25

and 75th quantiles are also called the quartiles of the probability law, since they cut the density into four pieces, each of which has area { under it. The interquartile range, t.15 - t 25, gives an interval, “centered” at the median, which includes 50 percent of the probability. It is sometimes used as an alternative measure of the variability of a probability law. The larger the difference, t15 - 12 , is, the larger will be the interval needed to con¬ tain 50 percent probability and thus, the more variable will be the observed values. The following example compares the median, mean, interquartile range, and standard deviation for a random variable. 5

example

3.3.3.

Suppose X is a random variable with distribution

function

Fx(t) =0,

t < 0

= y/t, = ,

< t < t > .

0

1

1

1

Let us find the median of X and the interquartile range for X. Since

Fx(t) = yfi, the median is defined by

\A.50 —

and thus 1

50

= (.5

)2

= .25. Similarly,

t = (.25)2 = .0625 t 15 = (,75 = .5625. 25

)2

so the interquartile range is t.15

t

25

— -5.

Let us also compute fix and ax. We find that the density function is

0

=

0,

< t