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Fundamentals of Mathematical Statistics (A Modern Approach), 10th Edition PDF

1303 Pages·2000·50.31 MB·English
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FUNDAMENTALS OF MATHEMATICAL STATISTICS (A Modern Approach) A Textbook written completely on modern lines for Degree, Honours, Post-graduate Students of al/ Indian Universities and ~ndian Civil Services, Indian Statistical Service Examinations. (Contains, besides complete theory, more than 650 fully solved examples and more than 1,500 thought-provoking Problems with Answers, and Objective Type Questions) S.C. GUPTA V.K. KAPOOR Reader in Statistics Reader in Mathematics Hindu College, Shri Ram Coffege of Commerce University of Delhi University of Delhi Delhi Delhi Tenth Revised Edition (Greatly Improved) '. -. .. I> \0 .be s~ .J.'''.'1eit ;;' +0\ -..' 'I1-f1- ,~,- 6 ~ ~ lr '<t - 8 ""0 o~ 'S'. 2002' ~ SULTAN CHAND & SONS Educational Publishers New Delhi * First Edition: Sept. 1970 Tenth Revised Ec;lition : August 2000 Reprint.: 2002 * Price: Rs. 210.00 ISBN 81-7014-791-3 * Exclusive publication, distribution and promotion rights reserved with the Publishers. * Published by : Sultan Chand & Sons 23, Darya Ganj, New Delhi-11 0002 Phones: V77843, 3266105, 3281876 * Laser typeset by : T.P. Printed at:· New A.S. Offset Press Laxmi Nagar Delhi·92 DEDICATED TO OUR TEACHER H. C. PROFESSOR GUPTA WHO INITIATED THE TEACHING OF MATHEMATICAL STATISTICS AT THE UNIVERSITY OF DELHI PREFACE TO THE TENrH EDITION The book has been revised keeping in mind the comments and suggestions received from the readers. An attempt is made to eliminate the misprints/errors in the last edition. Further suggestions and criticism for the improvement of ~he book will be' most welcome and thankfully acknowledged. \ August 2000 S.c. GUPTA V.K. KAPOOR TO THE NINTH EDITION The book originally written twenty-four years ago has, during the intervening period, been revised and reprinted seve'ral times. The authors have, however, been thinking, for the last f({w years that the book needed not only a thorough revision but rather a complete rewriting. They now take great pleilsure in presenting to the readers the ninth completely revised and enlarged edition of the book. The subject matter in the whole book has been rewritten in the light of numerous criticisms and suggestions received from the users of the previous editions in-lndia and abroad. Some salient features of the new edition are: • The entire text, especially Chapter 5 (Random Variables), Chapter 6 (Mathematical Expectation), Chapters 7 and 8 (Theoretical Discrete and Continuous Distributions), Chapter 10 (Correlation and Regression), Chapter 15 (Theory of Estimation), has been restructured, rewritten and updated to cater to the revised syllabi of Indian universities, Indian Civil Services and various other competitive examinations. • During the course of rewriting, it has been specially borne in mind to retain all the basic features of the previous editions especially the simplicity of presentation, lucidity of style and analytical approach which have been appreciated by teachers and students all over India and abroad. • A number of typical problems have been added as solved examples in each chapter. These will enable 'the reader to have a better and thoughtful understanding of the basic. concepts of the theor.y and its various applications. • Several new topics have been added at appropriate places to make the treatment more comprehensive and complete. Some of the obvious ADDITIONS are: § 8·1.5 Triangular Distribution p. 8· i 0 to 8·12 § 8·8.3 Logistic Distribution p. 8·92 to 8·95 § 8·10 Rem¥ks 2, Convergence in Distributipn of Law p. 8·106 § 8·10.3. Remark 3, Relation between Central ~imit Theorem al?d Weak Law of Large Numbers p. 8·110 § 8·10.4 C;ramer's Theorem p . 8·111-8.112, 8·114-8·115 - Example 8.46 J § 8· 74 to Order Statistics - Theory, Illustrations and § 8· 74·6, Exercise Set p. 8· 736 to 8· 751 § 8· 75 'Truncated Distributions-with Illustrations p. 8·757 to 8·756 ~ § 70·6· 7 Derivation of Rank Correlation Formula for Tied Ranks p. 70·40-70·47 . § 70-7· 7 Lines of Rt:;gression-Derivation (Aliter) p. 70·50-70·57. Example 70·27 p. 70·55 § 7O · 70·2 Remark to § 10· 70 ·2 - Marginal Distributions of Bivariate Normal Distribution p. 70·88-70·90 Tlieorem 70·5, p. 70·86. and Theorem 70·6, p. 70."(37 on Bivariate Normal Distribution. Solved Examples 70·37, 70·32, PClges 70·96·70·97 on BVN Distribution. Theorem 73·5 Alternative Proof of Distribution of (X, S2) using m.g.f. p. 73· 79 to 73·27 § 73· 77 X2-Test for pooling of Probabilities (PJ. Test) p. 73·69 § 75·4· 7 Invariance property of Consistent Estimators-TheQrem 75·7, pp 75·3 § 75·4·2 Sufficient Conditions for C;on~istency-Theorem 75·2, p. 75·3 § 15·5·5 MVUE: Theorem 75·4, p. 75·72-73·73 § 75·7 Remark 7. Minimum Variance Bound (MVB), Estimator, p.75·24 § 75·7· 7 Conditions for the equality sign in Cramer·Rao (CR) Inequality, p. 75·25 to 75·27 § 75·8 CQmplete family of Distributions (with illustrations), p. 75·37 to 75·34 Theorem 75·10 (Blackwellisation), p ..1 5·36. Theorems 75·76 and 75·77 on MLE, p. 75·55. § 76·5· 7 Unbiased Test and Unbiased Critical Region. Theorem 76·2·pages 76·9-76· 70 § 76·5·2 Optimum Regions and Sufficient Statistics, p.76·70-76·77 Remark to Example 76·6, p. 76· 77 · 76 · 78 and Remarks 7, 2 ,to Example 76·7, p. 76·20 to 76·22; GrqphicaI Representation of Critical Regions. • Exercise sets at the end of each chapter are substantially reorganised. Many new problems are included in the exercise sets. Repetition of questions of the same type (more than what is necessary) has been avoided. Further in the set of exercises, the problems have been carefully arranged and properly graded. More difficult problems are put in the miscellaneous exercise at the end of each chapter. • Solved examples and unsolved problems in the exercise sets 11Cfve been drav.:n from the latest examination papers of various Indian Universities, Indian' Civil Services, etc. (I'U) • An attempt has been made to rectify the errors in ·the previous editions . • The present edition Incorporates modern viewpoints. In fact with the addition of new topics, rewriting and revision of many others and restructuring of exercise sets, altogether a new book, covering the revised syllabi of almost all the Indian urilversities, is being presfJnted to the reader. It Is earnestly hoped that, In -the new form, the book will prove of much greater utility to the students as well as teachers of the subject. We express our deep sense of gratitude to our Publishers Mis sultan Chand & Sons and printers DRO Phototypesetter for their untiring efforts, unfailing courtesy, and co-operation in bringing out the book, in suchan elegant form. We are· also thankful to ou; several colleagues, friends and students for their suggestions and encouragement during the preparing of this revised edition; Suggestions and criticism for further improvement of the' book as weJl ~s intimation of errors and misprints will be most gratefully received and duly acknowledged. , S C. GUPTA & V.K. KAPOOR TO THE FIRST EDITION Although there are a iarge number of books available covering various aspects in the field of Mathematical Statistics, there is no comprehensive book dealing with the various topics on Mathematical Statistics for the students. The present book is a modest though detarmined bid to meet the requirf3ments of the students of Mathematical Statistics at Degree, Honours· and Post-graduate levels. The book will also be found' of use DY the students preparing for various competitive examinations. While writing this book our goal has been to present a clear, interesting, systematic and thoroughly teachable treatment of Mathematical Stalistics and to provide a textbook which should not only serve as an introduction to the study of Mathematical StatIstics but also carry the student on to 'such a level that he can read' with profit the numercus special monographs which are available on the subject. In any branch of Mathematics, it is certainly the teacher who holds the key to successful learning, Our aim in writing this book has been simply to assisf the teacher in conveying to th~ stude,nts .more effectively a thorough understanding of Mathematical Stat;st(cs. The book contains sixteen chapters (equally divided between two volumes). the first chapter is devoted to a concise and logical development of the subject. i'he second and third chapters deal with the frequency distributions, and measures of average, ~nd dispersion. Mathematical treatment has been given to .the proofs of various articles included in these chapters in a very logi9aland simple manner. The theory of probability which has been developed by the application of the set theory has been discussed quite in detail. A ,large number of theorems have been deduced using the simple tools of set theory. The (viii) simple applications of probability are also given. The chapters on mathematical expectation and theoretical distributions (discrete as well as continuous) have been written keeping the'latest ideas in mind. A new treatment has been given to the chapters on correlation, regress~on and bivariate normal distribution using the concepts of mathematical expectation. The thirteenth and fourteenth chapters deal mainly with the various sampling distributions and the various tests of significance which can be derived from them. In chapter 15, we have discussed concisely statistical inference (estimation and testing of hypothesis). Abundant material is given in the chapter on finite differences and numerical integration. The whole of the relevant theory is arranged in the form of serialised articles which are concise and to the. point without being insufficient. The more difficult sections will, in general, ,be found towards the end of each chapter. We have tried our best to present the subject so as to be within the easy grasp of students with vary~ng degrees of intellectual attainment. Due care has been taken of the examination r.eeds of the students and, wherever possible, indication of the year, when the' articles and problems were S!3t in the examination as been given. While writing this text, we have gone through the syllabi and examination papfJrs O,f almC'st all Inc;lian universities where the subject is taught sQ as to make it as comprehensive as possible. Each chapte( contains a large number of carefully graded worked problems mostly drawn from university papers with a view to acquainting the student with the typical questions pertaining to each topiC. Furthermore, to assist the student to gain proficiency iii the subject, a large number of properly graded problems maif)ly drawn from examination papers of various. universities are given at the end of each chapter. The questions and pro.blems given at the end of each chapter usually require for (heir solution a thoughtful use of concepts. During the preparation of the text we have gone through a vast body of liter9ture available on the subject, a list of which is given at the end of the book. It is expected that the bibliography given at the end of the book ,will considerably help those who want to make a detailed study of the subject • The lucidity of style and simplicity of expression have been our twin objects to remove the awe which is usually associated with most mathematical and statistical textbooks. While every effort has been made to avoid printing and other mistakes, we crave for the indulgence of the readers fot the errors that might have inadvertently crept in. We shall consider our efforts amplY rewarded if those for whom the book is intended are benefited by it. Suggestions for the improvement of the book will be hIghly appreciated and will be duly incorporated. SEPTEMBER 10, 1970 S.C. GUPTA & V.K. KAPOOR contents Pages ~rt Chapter 1 Introduction -- Meaning and Scope 1-1 - 1'8 1 '1 Origin and Development of Statistics 1-1 1'2 Definition of Statistics 1'2 1'3 Importance and Scope of Statistics 1-4 1'4 Limitations of Statistics 1-5 1·5 Distrust of Statistics 1-6 Chapter 2 Frequency Distributions and Measures of central Tendency .2-1 - 2·44 2'1 Frequency Distribution$ 2·1 2·1'1 Continuous Frequency ,Distribution 2-4 2-2 Graphic Representation of a Frequency Distribution 2-4 2-2-1 Histogram 2-4 2'2'2 Frequency Polygon 2·5 2'3 Averages or Measures of Central Tendency or Measures of Location 2'6 2'4 Requisites for an Ideal Measure of Central Tendency 2-6 2·5 Arithmetic Mean 2-6 2·5'1 Properties of Arithmetic Mean 2·8 2·5'2 Merits and Demerits of Arithmetic Mean 2'10 2-5'3 Weighted Mean 2-11 2'6 Median 2'13 2·6'1 Derivation of Median Formula 2'19 2-6'2 Merits and Demerits of Median 2·16 2-7 Mode 2'17 2-7-1 Derivation of Mode Formula 2'19 2·7'2 Merits and Demerits of Mode 2-22 2'8 Geometric Mean 2'22 2'8-1 Merits and Demerits of Geometric Mean 2'23 2'9 Harmonic Mean 2-25 2'9'1 Merits and Demerits of Harmonic Mean 2'25 2~1 0 Selection of an Average 2'26 2'11 Partition Values 2'26 / 2'11'1 Graphicai Location of the Partition Values 2'27 (x) Chapter-3 Measures of Dispersion, Skewness and Kurtosis 3'1 - 3·40 3·1 Dispersion 3'1 3-2 Characteristics for an Ideal Measure of DisperSion 3-1 3·3 Measures of Dispersion 3-1 3·4 Range 3-1 3:~ Quartile Deviation 3-1 3'6 Mean Deviation 3-2 3·7 Standard Deviation (0) and,Root Mean Square Deviation (5) 3'2 3·7·1 Relation between 0 and s' 3'3 3·7·2 Different Formulae for Calculating Variance. 3'3 3·7'3 Variance of the Combined Series 3'10 3-8 Coefficient of Dispersion 3'12 3'8·1 Coefficient of Variation 3'12 3-9 Moments 3'21 3'9'1 Relation Between Moments About Mean in Terms of Moments About Any Point and Vice Versa 3'22 3'9-2 Effect of Cllange of Origin and Scale on Moments 3-23 3'9·3' Sheppard's Correction for Moments 3-23 3-9-4 Charlier's Checks 3'24 3'10' Pearson's ~ and y Coefficients 3'24 3'11 Factorial Moments 3-24 3'12 Absolute Moments • 3-25 3'13 Skewness 3-32 3'14 Kurtosis 3-35 Chapter- 4 Theory of Probability 4-1 - 4·116 4'1 Introduction 4-1 4·2 S~ort History 4-1 4·3 Definitions of Various Terms 4-2 4'3-1 Mathematical or Classical Probability 4-3 4-3-2 Statistical or Empirical Probability 4-4 4-4 Mathematicallools : Preliminary Notions of Sets 4-14 4-4-1 Sets and Elements of Sets 4'14 .4-4-2 Operations on Sets 4-15 4;4-3 Albebra of Sets 4-15 4-4-4 Umn of. Sequence of Sets 4-16 4'4·5 Classes of S~ts 4-17 4-5 Axiomatic ApP,roach to Probability 4'17

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