Table Of ContentFUNDAMENTALS 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)
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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
