Schaum's Outline of Probability and Statistics, 3rd Ed. (Schaum's Outline Series)

433 Pages·2008·9 MB·English

by  John Schiller

Probability and
Statistics
Third Edition
Murray R. Spiegel, PhD
Former Professor and Chairman of Mathematics
Rensselaer Polytechnic Institute
Hartford Graduate Center
John J. Schiller, PhD
Associate Professor of Mathematics
Temple University
R. Alu Srinivasan, PhD
Professor of Mathematics
Temple University
Schaum’s Outline Series
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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 con-
temporary 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 ac-
ceptance 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 Statisticsby 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 cu-
mulative distribution function since it is a more fundamental concept than the probability density function. Also,
more emphasis is placed on the Pvalues of hypotheses tests,since technology has made it possible to easily de-
termine 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 prob-
ability and statistics using a background of calculus—and the features that made the first edition such a great suc-
cess have been preserved,and we hope that this edition can serve an even broader range of students.
J. SCHILLER
R. A. SRINIVASAN
iii
Preface to the
First Edition
The important and fascinating subject of probability began in the seventeenth century through efforts of such math-
ematicians 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 math-
ematics 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,organ-
ization,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 ref-
erence 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 one-semester 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,Agri-
cultural 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
iv
Contents
Part I PROBABILITY 1
CHAPTER 1 Basic Probability 3
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 Approxima-
tion to n!
CHAPTER 2 Random Variables and Probability Distributions 34
Random Variables Discrete Probability Distributions Distribution Functions for Random
Variables Distribution Functions for Discrete Random Variables Continuous Random Vari-
ables Graphical Interpretations Joint Distributions Independent Random Variables
Change of Variables Probability Distributions of Functions of Random Variables Convo-
lutions Conditional Distributions Applications to Geometric Probability
CHAPTER 3 Mathematical Expectation 75
Definition of Mathematical Expectation Functions of Random Variables Some Theorems
on Expectation The Variance and Standard Deviation Some Theorems on Variance Stan-
dardized Random Variables Moments Moment Generating Functions Some Theorems
on Moment Generating Functions Characteristic Functions Variance for Joint Distribu-
tions. 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 4 Special Probability Distributions 108
The Binomial Distribution Some Properties of the Binomial Distribution The Law of
Large Numbers for Bernoulli Trials The Normal Distribution Some Properties of the Nor-
mal Distribution Relation Between Binomial and Normal Distributions The Poisson Dis-
tribution 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 The Cauchy Distribution The Gamma Distribution The Beta
Distribution The Chi-Square Distribution Student’s tDistribution The FDistribution
Relationships Among Chi-Square,t,and FDistributions The Bivariate Normal Distribution
Miscellaneous Distributions
v
vi Contents
Part II STATISTICS 151
CHAPTER 5 Sampling Theory 153
Population and Sample. Statistical Inference Sampling With and Without Replacement
Random Samples.Random Numbers Population Parameters Sample Statistics
Sampling Distributions The Sample Mean Sampling Distribution of Means Sampling
Distribution of Proportions Sampling Distribution of Differences and Sums The Sample
Variance Sampling Distribution of Variances Case Where Population Variance Is Un-
known Sampling Distribution of Ratios of Variances Other Statistics Frequency Distri-
butions Relative Frequency Distributions Computation of Mean,Variance,and Moments
for Grouped Data
CHAPTER 6 Estimation Theory 195
Unbiased Estimates and Efficient Estimates Point Estimates and Interval Estimates. Relia-
bility Confidence Interval Estimates of Population Parameters Confidence Intervals for
Means Confidence Intervals for Proportions Confidence Intervals for Differences and
Sums Confidence Intervals for the Variance of a Normal Distribution Confidence Intervals
for Variance Ratios Maximum Likelihood Estimates
CHAPTER 7 Tests of Hypotheses and Significance 213
Statistical Decisions Statistical Hypotheses. Null Hypotheses Tests of Hypotheses and
Significance Type I and Type II Errors Level of Significance Tests Involving the Normal
Distribution One-Tailed and Two-Tailed Tests PValue Special Tests of Significance for
Large Samples Special Tests of Significance for Small Samples Relationship Between
Estimation Theory and Hypothesis Testing Operating Characteristic Curves. Power of a Test
Quality Control Charts Fitting Theoretical Distributions to Sample Frequency Distributions
The Chi-Square Test for Goodness of Fit Contingency Tables Yates’Correction for Con-
tinuity Coefficient of Contingency
CHAPTER 8 Curve Fitting,Regression,and Correlation 265
Curve Fitting Regression The Method of Least Squares The Least-Squares Line The
Least-Squares Line in Terms of Sample Variances and Covariance The Least-Squares
Parabola Multiple Regression Standard Error of Estimate The Linear Correlation
Coefficient Generalized Correlation Coefficient Rank Correlation Probability Interpreta-
tion of Regression Probability Interpretation of Correlation Sampling Theory of Regression
Sampling Theory of Correlation Correlation and Dependence
CHAPTER 9 Analysis of Variance 314
The Purpose of Analysis of Variance One-Way Classification or One-Factor Experiments
Total Variation. Variation Within Treatments. Variation Between Treatments Shortcut Meth-
ods for Obtaining Variations Linear Mathematical Model for Analysis of Variance Ex-
pected Values of the Variations Distributions of the Variations The F Test for the Null
Hypothesis of Equal Means Analysis of Variance Tables Modifications for Unequal Num-
bers of Observations Two-Way Classification or Two-Factor Experiments Notation for
Two-Factor Experiments Variations for Two-Factor Experiments Analysis of Variance for
Two-Factor Experiments Two-Factor Experiments with Replication Experimental Design
vii
Contents
CHAPTER 10 Nonparametric Tests 348
Introduction The Sign Test The Mann–Whitney UTest The Kruskal–Wallis HTest
TheHTest Corrected for Ties The Runs Test for Randomness Further Applications of
the Runs Test Spearman’s Rank Correlation
CHAPTER 11 Bayesian Methods 372
Subjective Probability Prior and Posterior Distributions Sampling From a Binomial Pop-
ulation Sampling From a Poisson Population Sampling From a Normal Population with
Known Variance Improper Prior Distributions Conjugate Prior Distributions Bayesian
Point Estimation Bayesian Interval Estimation Bayesian Hypothesis Tests Bayes Fac-
tors Bayesian Predictive Distributions
APPENDIX A Mathematical Topics 411
Special Sums Euler’s Formulas The Gamma Function The Beta Function Special
Integrals
APPENDIX B Ordinates yof the Standard Normal Curve at z 413
APPENDIX C Areas under the Standard Normal Curve from 0 to z 414
APPENDIX D Percentile Values t for Student’s tDistribution
p
with nDegrees of Freedom 415
APPENDIX E Percentile Values x2 for the Chi-Square Distribution
p
with n Degrees of Freedom 416
APPENDIX F 95th and 99th Percentile Values for the FDistribution
with n ,n Degrees of Freedom 417
1 2
APPENDIX G Values of e2l 419
APPENDIX H Random Numbers 419
SUBJECT INDEX 420
INDEX FOR SOLVED PROBLEMS 423
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PART I
Probability