The late MURRAY R SPIEGEL received an MS degree in physics and a PhD in mathematics from Cornell University He had positions at Harvard University, Columbia University, Oak Ridge, and Rensselaer Polytechnic Institute and served as a mathematical consultant at several large companies His last position was professor and chairman of mathematics at Rensselaer Polytechnic Institute, Hartford Graduate Center He was interested in most branches of mathematics, especially those which involve applications to physics and engineering problems He was the author of numerous journal articles and 14 books on various topics in mathematics JOHN J SCHILLER is an associate professor of mathematics at Temple University He received his PhD at the University of Pennsylvania He has published research papers in the areas of Riemann surfaces, discrete mathematics, and mathematical biology He has also coauthored texts in finite mathematics, precalculus, and calculus R ALU SRINIVASAN is a professor of mathematics at Temple University He received his PhD at Wayne State University and has published extensively in probability and statistics Copyright © 2013 by The McGraw-Hill Companies, Inc All rights reserved Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher ISBN: 978-0-07-179558-6 MHID: 0-07-179558-8 The material in this eBook also appears in the print version of this title: ISBN: 978-0-07-179557-9, MHID: 0-07-179557-X All trademarks are trademarks of their respective owners Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark Where such designations appear in this book, they have been printed with initial caps McGraw-Hill eBooks 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derivative works based upon, transmit, distribute, disseminate, sell, publish or sublicense the work or any part of it without McGraw-Hill’s prior consent You may use the work for your own noncommercial and personal use; any other use of the work is strictly prohibited Your right to use the work may be terminated if you fail to comply with these terms THE WORK IS PROVIDED “AS IS.” McGRAW-HILL AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING ANY INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE McGraw-Hill and its licensors not warrant or guarantee that the functions contained in the work will meet your requirements or that its operation will be uninterrupted or error free Neither McGraw-Hill nor its licensors shall be liable to you or anyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any damages resulting therefrom McGraw-Hill has no responsibility for the content of any information accessed through the work Under no circumstances shall McGraw-Hill and/or its licensors be liable for any indirect, incidental, special, punitive, consequential or similar damages that result from the use of or inability to use the work, even if any of them has been advised of the possibility of such damages This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise Preface to the Third Edition In the second edition of Probability and Statistics, which appeared in 2000, the guiding principle was to make changes in the first edition only where necessary to bring the work in line with the emphasis on topics in contemporary texts In addition to refinements throughout the text, a chapter on nonparametric statistics was added to extend the applicability of the text without raising its level This theme is continued in the third edition in which the book has been reformatted and a chapter on Bayesian methods has been added In recent years, the Bayesian paradigm has come to enjoy increased popularity and impact in such areas as economics, environmental science, medicine, and finance Since Bayesian statistical analysis is highly computational, it is gaining even wider acceptance with advances in computer technology We feel that an introduction to the basic principles of Bayesian data analysis is therefore in order and is consistent with Professor Murray R Spiegel’s main purpose in writing the original text—to present a modern introduction to probability and statistics using a background of calculus J SCHILLER R A SRINIVASAN Preface to the Second Edition The first edition of Schaum’s Probability and Statistics by Murray R Spiegel appeared in 1975, and it has gone through 21 printings since then Its close cousin, Schaum’s Statistics by the same author, was described as the clearest introduction to statistics in print by Gian-Carlo Rota in his book Indiscrete Thoughts So it was with a degree of reverence and some caution that we undertook this revision Our guiding principle was to make changes only where necessary to bring the text in line with the emphasis of topics in contemporary texts The extensive treatment of sets, standard introductory material in texts of the 1960s and early 1970s, is considerably reduced The definition of a continuous random variable is now the standard one, and more emphasis is placed on the cumulative distribution function since it is a more fundamental concept than the probability density function Also, more emphasis is placed on the P values of hypotheses tests, since technology has made it possible to easily determine these values, which provide more specific information than whether or not tests meet a prespecified level of significance Technology has also made it possible to eliminate logarithmic tables A chapter on nonpara-metric statistics has been added to extend the applicability of the text without raising its level Some problem sets have been trimmed, but mostly in cases that called for proofs of theorems for which no hints or help of any kind was given Overall we believe that the main purpose of the first edition—to present a modern introduction to probability and statistics using a background of calculus—and the features that made the first edition such a great success have been preserved, and we hope that this edition can serve an even broader range of students J SCHILLER R A SRINIVASAN Preface to the First Edition The important and fascinating subject of probability began in the seventeenth century through efforts of such mathematicians as Fermat and Pascal to answer questions concerning games of chance It was not until the twentieth century that a rigorous mathematical theory based on axioms, definitions, and theorems was developed As time progressed, probability theory found its way into many applications, not only in engineering, science, and mathematics but in fields ranging from actuarial science, agriculture, and business to medicine and psychology In many instances the applications themselves contributed to the further development of the theory The subject of statistics originated much earlier than probability and dealt mainly with the collection, organization, and presentation of data in tables and charts With the advent of probability it was realized that statistics could be used in drawing valid conclusions and making reasonable decisions on the basis of analysis of data, such as in sampling theory and prediction or forecasting The purpose of this book is to present a modern introduction to probability and statistics using a background of calculus For convenience the book is divided into two parts The first deals with probability (and by itself can be used to provide an introduction to the subject), while the second deals with statistics The book is designed to be used either as a textbook for a formal course in probability and statistics or as a comprehensive supplement to all current standard texts It should also be of considerable value as a book of reference for research workers or to those interested in the field for self-study The book can be used for a one-year course, or by a judicious choice of topics, a onesemester course I am grateful to the Literary Executor of the late Sir Ronald A Fisher, F.R.S., to Dr Frank Yates, F.R.S., and to Longman Group Ltd., London, for permission to use Table III from their book Statistical Tables for Biological, Agricultural and Medical Research (6th edition, 1974) I also wish to take this opportunity to thank David Beckwith for his outstanding editing and Nicola Monti for his able artwork M R SPIEGEL Contents Part I PROBABILITY CHAPTER Basic Probability Random Experiments Sample Spaces Events The Concept of Probability The Axioms of Probability Some Important Theorems on Probability Assignment of Probabilities Conditional Probability Theorems on Conditional Probability Independent Events Bayes’ Theorem or Rule Combinatorial Analysis Fundamental Principle of Counting Tree Diagrams Permutations Combinations Binomial Coefficients Stirling’s Approximation to n! CHAPTER Random Variables and Probability Distributions Random Variables Discrete Probability Distributions Distribution Functions for Random Variables Distribution Functions for Discrete Random Variables Continuous Random Variables Graphical Interpretations Joint Distributions Independent Random Variables Change of Variables Probability Distributions of Functions of Random Variables Convolutions Conditional Distributions Applications to Geometric Probability CHAPTER Mathematical Expectation Definition of Mathematical Expectation Functions of Random Variables Some Theorems on Expectation The Variance and Standard Deviation Some Theorems on Variance Standardized Random Variables Moments Moment Generating Functions Some Theorems on Moment Generating Functions Characteristic Functions Variance for Joint Distributions Covariance Correlation Coefficient Conditional Expectation, Variance, and Moments Chebyshev’s Inequality Law of Large Numbers Other Measures of Central Tendency Percentiles Other Measures of Dispersion Skewness and Kurtosis CHAPTER Special Probability Distributions The Binomial Distribution Some Properties of the Binomial Distribution The Law of Large Numbers for Bernoulli Trials The Normal Distribution Some Properties of the Normal Distribution Relation Between Binomial and Normal Distributions The Poisson Distribution Some Properties of the Poisson Distribution Relation Between the Binomial and Poisson Distributions Relation Between the Poisson and Normal Distributions The Central Limit Theorem The Multinomial Distribution The Hypergeometric Distribution The Uniform Distribution Permutations, 9, 18, 28 Point estimate, 195 Poisson distribution, 111, 128, 129, 146 Pooled variance, 218 Population, 153 correlation coefficient, 273 parameters, 154 size, 153 Posterior probability, 373 Power of a test, 219, 235 Precision of a distribution, 377 Predictive intervals, 388 Predictive point estimate, 388 Prior and posterior distributions, 372, 388 when sampling from a binomial population, 375, 390 when sampling from a Poisson population, 376, 391 when sampling from a normal population, 377, 391 Prior and posterior odds ratios, 384 Prior probability, 373 Probability, axioms of, calculation, 12, 22, 23 classical approach, concept of, density function, 37 discrete, 34 distribution, 34, 37, 44 distributions of functions of random variables, 42 frequency approach, function, 34 graph paper, 219 interpretation of correlation, 273, 295 interpretation of regression, 271, 295 joint, 39, 47, 48 of an event, of failure, 108 of success, 108 surface, 40 using combinatorial analysis, 22 Product-moment formula, 270, 290 Quadratic curve, 265 Quality control charts, 219, 238 Random experiments, 3, 10 Random numbers, 154 Random sample, 154 Random variable, 34 continuous, 36 discrete, 34, 44 nondiscrete, 34 Randomized blocks, 323 Rank correlation, 271, 293, 365 Region of: acceptance, 214 nonsignificance, 214 rejection, 214 significance, 214 Regression, 265 equation, 265, 269 line, 272 plane, 269 surface, 269 Relationship: among, chi-square, F, and t distributions, 117, 139 between binomial and normal distributions, 111 between binomial and Poisson distributions, 111 between Poisson and normal distributions, 112 between estimation theory and hypothesis testing, 219 Relative (percentage) frequency distribution, 161 Reliability, 195 Repetitions, 314 Replications, 314, 321 Residual (in regression), 266 Residual variation, 319 Row means, 314 rth moment: about the mean, 78–79 about the origin, 82 central, 78–79 conditional, 82 raw, 79 Run, 351 Runs test for randomness, 350, 361, 364 Sample, 153 Sample correlation coefficient, 268 Sample mean, 155, 163, 177, 178 Sample moments, 177 Sample point, Sample size, 153 Sample space, 3, 10 countably infinite, discrete, finite, Sample statistic, 154 Sample variance, 157, 177 Sampling, 153 Sampling distribution of means, 155, 163, 181 related theorems when population variance is known, 155–156 when population variance is not known, 159, 174 Sampling distribution, 155, 163, 166, 169, 171 of differences and sums, 157, 169 of proportions, 156, 166 of ratios of variances, 159, 174 of variances, 158, 169, 171 Sampling theory of correlation, 274, 298 Sampling theory of regression, 297 Sampling with replacement, 112, 153 Sampling without replacement, 113, 153 Scatter diagram, 265, 280 Semi-interquartile range, 84 Sign test, 348, 352 Significance of a difference between correlation coefficients, 274 Skewness, 84, 96, 97, 181 Spearman’s formula for rank correlation, 352 Spearman’s rank correlation coefficient, 271 Standard deviation, 77, 87 Standard error, 155 table for, 160 Standard error of estimates, 269, 287 Standard normal curve, 110 Standard normal density function, 110 Standard score, 78, 110 Standard units, 78 Standardized random variable, 78 Statistic, 154, 155 Statistical decisions, 213 Statistical hypotheses, 213 Statistical inference, 153 Statistical significance, 214 Stirling’s formula, 10 Strong law of large numbers, 83 Student’s t distribution, 115, 136 tests involving, 236 Subjective probability, 372, 388 t distribution, 115 tests involving, 236 Tests of hypothesis and significance, 213–264 for correlation coefficient, 274 for differences between correlation coefficients, 274 for differences of means, 217, 218, 227, 255 for differences of proportions, 217, 227, 253 involving chi-square distribution, 217, 233, 242, 257 involving F distribution, 217, 233, 257 involving the normal distribution, 214, 222, 255 involving Student’s t distribution, 218, 230 for large samples, 216, 217 for means, 216, 217, 222, 255 for predicted values in linear regression, 273 for proportions, 216, 222, 255 for ratios of variances, 218 for regression coefficient in linear regression, 273 for small samples, 217 for variances, 218 Theoretical frequency, 221 Theory of runs, 350 Three-factor experiments, 329 Three-way classification, 329 Total variation, 270, 273, 315, 319 Transformed variables, 265 Translation of axes, 267 Treatment effects, 320 Treatment means, 314 Treatments, 314 Treatments and blocks, 318 Tree diagram, 8, 17 Trend pattern, 351, 362 Trend values, 301 Two-factor experiments with replications, 321, 331 Two-factor experiments, 318, 330 Two-sided tests, 214 Two-tailed tests, 214 Two-way classification, 318, 330 Type I error, 213 Type II error, 213 Unbiased estimate, 158, 195, 199 Unbiased estimator, 158, 195, 199 Unequal numbers of observations, 318 Unexplained variation, 270, 273 Uniform distribution, 113–114, 132, 133 Vague prior distribution, 373 Variance, 77, 78, 81, 87, 100 for grouped data, 161, 162 for samples, 177 of binomial distribution, 109 conditional, 82, 93, 102 of F distribution, 116 of normal distribution, 110 of Student’s t distribution, 116 pooled, 218 sampling distribution of, 171 Variation: between treatments, 315 expected value, 316 explained, 289 distribution, 317 for two-factor experiments, 319 residual, 319 shortcut methods for obtaining, 315 total, 289, 315 unexplained, 292 within treatments, 315 Weak law of large numbers, 83 Weibull distribution, 118, 141 Yates’s correction for continuity, 221, 242, 244, 247 Z transformation, Fischer’s, 274 Index for Solved Problems Please note that index links point to page beginnings from the print edition Locations are approximate in e-readers, and you may need to page down one or more times after clicking a link to get to the indexed material Bayes factor, 400 Bayesian: hypothesis tests, 399 interval estimation, 397 point estimation, 394 predictive distributions, 401 Bayes’s theorem, 17 Beta distribution, 133 Binomial coefficients, 21 Binomial distribution, 118 moment generating function of, 121 normal approximation to, 126, 129 Poisson approximation to, 128 Bivariate normal distribution, 140 Buffon’s needle problem, 64 Calculation of probabilities, 12 Cauchy distribution, 132, 133 characteristic function of, 132 moment generating function of, 132 relation to uniform distribution, 133 Central limit theorem, 129 for binomial random variables, 129 proof of, 130 Central tendency, measures of, 94 Change of variables, 51, 63 Characteristic function, 90, 97 of Cauchy distribution, 132 Chebyshev’s inequality, 93 Chi-square distribution: moment generating function of, 134 relationship to F and t distributions, 139 relationship to normal distribution, 134 tests involving, 233, 242 Chi-square test of goodness of fit, 242, 246, 252 Coefficient of: contingency, 250 correlation, 91 determination, 301 linear correlation, 289 Combinations, 20 Combinatorial analysis, 17 probability using, 22 Conditional: density, 58 distribution, 58 expectation, 93 moments, 93 probability, 14 variance, 93 Confidence interval estimates for: differences of means, 203 differences of proportions, 202 means in large samples, 200 means in small samples, 202 mean when population variance is unknown, 174 proportions, 202, 207 standard deviation, 208 variances, 204 variance ratios, 205 Conjugate prior distributions, 393 Contingency, coefficient of, 250 Contingency tables, 246 Continuous distribution function, 46 Convolutions, 56 Correlation: coefficient, 91, 140 generalized, 292 linear, 289 multiple, 293 probability interpretation of, 295 product-moment formula for, 290 rank, 293 sampling theory of, 298 table, 305 Counting, 17 Covariance, 91, 184 Cyclic pattern, in runs test, 361 Determination, coefficient of, 301 Discrete distribution function, 45 Discrete random variable, 44 Dispersion, measures of, 96 Distribution: Bayesian predictive, 401 beta, 133 binomial, 118 bivariate normal, 140 Cauchy, 132 chi-square, 134 conditional, 58 conjugate prior, 393 continuous, 46 of differences and sums, 169 discrete, Fisher’s F, 138 frequency, 175 gamma, 133 hypergeometric, 131 improper prior, 392 joint, 47 marginal, 48 multinomial, 131 normal, 122 of means, 163 of proportions, 166 of ratios of variances, 174 Poisson, 128 prior and posterior, 388 relationships among, F, chi-square, and t, 139 sampling, 163, 166, 169, 171, 174 Student’s t, 136 uniform, 132 of variances, 171 Weibull, 141 Distribution functions: continuous, 46 discrete, 45 Marginal, 48, 49 Efficient estimates, 199 Estimates: confidence interval, 200, 202 efficient, 199 maximum likelihood, 206 unbiased, 199 Events, 10 independent, 14 mutually exclusive, 13 Expectation of random variables, 85 conditional, 93 F distribution, 138 relationship to chi-square and t distributions, 139 tests involving, 233 Fitting of data by theoretical distributions, 239 Fourier coefficients, 97 Fourier series, 97 Frequency: distributions, 175 histograms, 176 polygons, 176 Gamma distribution, 133 Generalized correlation coefficient, 292 Geometric probability, applications to, 60 Goodness of fit, 242, 246 Graeco-Latin squares, 335 Hypergeometric distribution, 131 Hypotheses tests (see Tests) Improper distributions, 392 Independent events, 14 Independent random variables, 47, 59 Joint density functions, 48 Joint distributions, 47 Kruskal-Wallis H test, 283 Kurtosis, 96, 97, 98 Latin squares, 334 Law of large numbers, 93 for Bernoulli trials, 122 Least squares: line, 275 parabola, 284 Linear correlation coefficient (see Correlation) Linear relationship, 292 Log-log graph paper, 283 Mann-Whitney U test, 354 sampling distribution of, 358 Marginal density function, 58 Marginal distribution function, 48, 61 Maximum likelihood estimates, 206 for mean of normal distribution, 206 for variance of normal distribution, 207 Means: computation of, for samples, 177 sampling distribution of, 163 Measures of central tendency, 94 Measures of dispersion, 96 Median, 94 Mode, 94 Modifications for unequal number of observations, 328 Moment generating function, 88, 96 of binomial distribution, 121 of Cauchy distribution, 132 of Poisson distribution, 129 of sums of independent random variables, 89 Moments, 88 computation of, 93 conditional, 93 for samples, 177 Multinomial distribution, 131 Multiple correlation coefficient, 293 Mutually exclusive events, 13 Nonlinear equations reducible to linear equations, 282 Nonlinear relationship, 292 Normal approximation to binomial distribution, 126, 129 Normal distribution, 122 bivariate, 140 One-factor experiments, 324 One-way classification, 324 Operating characteristic curve, 234, 251 Operating characteristic function, 235 Percentiles, 96 Permutations, 18 Poisson distribution, 128 moment generating function of, 129 Power function, 235 Prior and posterior distributions, 388 when sampling from a binomial population, 390 when sampling from a normal population, 391 when sampling from a Poisson population, 391 Probability: calculation of, 12 calculating using combinatorial analysis, 22, 23 conditional, 14 distributions, 44 geometric, 60 subjective, 388 theorems on, 11 Probability distributions: continuous, 46 discrete, 44 Probability interpretation of correlation and regression, 295 Product-moment formula, 290 Proportions, sampling distribution of, 166 Quality control charts, 238 Random experiments, 10 Random variables: conditional expectation of, 93 continuous, 46 discrete, 44 expectation of, 85 independent, 47 Randomness runs test, 361 Rank correlation, 293, 365 Regression: least-squares line of, 275 multiple, 285 probability interpretation of, 295 sampling theory of, 297 Runs test for randomness, 361 applications of, 364 Sample mean: coding formula for, 178 computation of, 177 Sample moments, computation of, 177 Sample spaces, 10 Sample variance: coding formula for, 179 computation of, 177 Sampling distribution of: difference of means, 169 mean when population variance is unknown, 174 means, 163, 181 proportions, 166 ratios of variances, 174 sum of variances, 169 variances, 171 Sampling theory of: correlation, 298 regression, 297 Scatter diagram, 280 Sign test, 352 Skewness, 96, 97, 181 Standard deviation, 87 Standard error of estimate, 287 Student’s t distribution, 136 relationship to F and chi-square distributions, 139 tests involving, 236 Subjective probability, 388 Tests: of difference of means, 227 of differences of proportions, 227 involving the chi-square distribution, 233, 242 involving the F distribution, 233 involving the Student’s t distribution, 230 of means using normal distributions, 222 of proportions using normal distributions, 222 Theoretical distributions, fitting of data by, 239 Three-factor experiments, 329 Three-way classification, 329 Tree diagrams, 17 Trend pattern, in runs test, 362 Trend values, 301 Two-factor experiments, 330 with replication, 331 Two-way classification, 330 Unbiased estimates, 199 Uniform distribution, 132 Variables, change of, 51, 63 Variables, random (see Random variables) Variance, 87 computation of, for samples, 177 conditional, 93 sampling distribution of, 171 Variation: explained, 289 total, 289 unexplained, 292 Weibull distribution, 141 ... Second Edition The first edition of Schaum’s Probability and Statistics by Murray R Spiegel appeared in 1975, and it has gone through 21 printings since then Its close cousin, Schaum’s Statistics. .. first edition to present a modern introduction to probability and statistics using a background of calculus and the features that made the first edition such a great success have been preserved, and. .. CHAPTER Random Variables and Probability Distributions Random Variables Discrete Probability Distributions Distribution Functions for Random Variables Distribution Functions for Discrete Random