Probability and Statistics

Transcription

1 Probability and Statistics

2 Probability and Statistics A.M. Mathai Associate Professor of Mathematics and Statistics McGill University, A-fontreal, Canada P. N. Rathie Professor of Statistics, Instituto de Matematicd e Estatistica, Universidade Estadual de Campinas, Brazil M

3 @A. M. Mathai and P. N. Rathie 1977 All rights reserved. No part of this publication may be reproduced or transmitted, in any form or by any means, without permission. First published in India 1977 by THE MACMILLAN COMPANY OF INDIA LIMITED Delhi Bombay Calcutta Madras First published in the United Kingdom 1977 by THE MACMILLAN PRESS LTD London and Basingstoke Associated companies in New rork Dublin Melbourne Johannesburg and Madras ISBN ISBN (ebook) DOI / This book is sold subject to the standard conditions of the Net Book Agreement. The paperback edition of this hook is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, resold, hired out, or otherwise circulated without the publisher's prior consent in any from of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser.

4 Preface This book is an introduction to basic statistical concepts and their applications. The only prerequisite is some knowledge of differential and integral calculus, which can be picked up in fifteen one-hour lectures. All other mathematical requirements are given in the book itself, in the form of technical notes. The book is intended to give a clear picture of basic statistical concepts. Since the theory of sets is usually taught in secondary schools, only the concepts needed for a meaningful discussion of basic probability theory and statistical populations are discussed; the utmost care is taken to make the treatment rigorous at the mathematical level assumed. Every concept is properly defined and illustrated with examples and counter examples. The important points are specifically made in ' comments ' following the examples. A sufficient number of problem for a complete understanding of the topics discussed is given at the end of each section, and an additional set of problems appears at the end of each chapter, some of which supplement the theory in that chapter. Answers to these problems are provided at the end of the book. Also a number of statistical decision problems are included following the discussion of mathematical expectation. Acceptance sampling and quality control problems are discussed after introducing probability models, with a view to giving some practical applications of the models. A brief introduction to the topic of entropy is also given. Other applications have been omitted to keep the book to a manageable size. All the important results, correspondences and relationships are given in tabular form for quick reference. Most of the illustrations are taken from daily life, keeping in mind the age group of students for whom the book is intended.

5 Vl PREFACE It is suggested that the book be used for a twenty-five-week course of three hours per week, given as a first course in statistics for arts, science, commerce and engineering students. This could usefully be supplemented by an additional one-hour problems session each week for those students who have difficulty with the subject. A.M. MATHAI P. N. RATHffi

6 Contents Preface v List of Symbols XlV Part I BAsra STATISTICAL CoNCEPTS Chapter 1 Statistical Populations and Sampling Sets A statistical population A subset A sample A representative sample Sampling techniques Random sampling numbers Simple random sampling Stratified sampling Representation of a numerical sample Classification Bar diagrams Histograms Frequency polygons and curves Ogives Pie diagrams Pictograms Technical notes (E and'" notations) 24

7 viii CONTENTS Measures of central tendency and dispersions Measures of central tendency The mean The Fractiles The mode Measures of dispersion The standard deviation The mean absolute deviation The range and the interquartile range Additional 52 Chapter 2 An OutcoiDe Set Random experiment The outcome set or the sure event An event Algebra of sets Union of sets Intersection of sets Mutually exclusive and impossible events Complements and non-occurrence of events A correspondence Venn diagrams Technical note (permutations and combinations) Permutations Combinations Stirling's approximation Binomial coefficients Multinomial coefficients 76 77

8 CONTENTS ix Additional 78 Chapter 3 Probability Probability of an event How to assign probabilities to various events The method of symmetry The method of relative frequencies Conditional probability Independence of events Pairwise and mutual independence Bayes' rule Entropy of a finite scheme A complete system of events A finite scheme Entropy Additional 104 Partll PROBABILITY MoDELs Chapter 4 Stochastic Variables Discrete and continuous stochastic variables A probability function II3 II The distribution function Mathematical expectation Moments Some decision problems The moment generating function

9 X CONTENTS AdditionaJ 140 Chapter 5 Univariate Probability Models-Discrete Discrete univariate models 5.2 The Binomial model 5.21 The mean value and the variance 5.3 The Poisson model 5.4 Other discrete models 5.5 Acceptance sampling 5.51 The operating characteristic curve 5.52 Producer's and consumer's risks Additional Chapter 6 Univariate Probability Models-Continuous Continuous univariate models Technical notes-gamma, Beta, Even and Odd functions The normal distributions The mean value and the variance A distribution-free property Other continuous models Change ofvariables Additional 198 Chapter 7 Sampling Distributions Independent stochastic variables A random sample from a theoretical population A statistic The Sampling distribution 213

10 CONTENTS 7.23 The central limit theorem 7.3 Sampling distributions associated with population 7.31 The distribution of the sample mean 7.32 The chi-square distribution 7.33 The Student-t distribution 7.34 The F-distribution 7.4 Control charts 7.41 Control charts for means 7.42 Control charts for proportions 7.5 Multivariate distributions 7.51 Change of variables 7.52 The multinomial distribution 7.53 Conditional distributions Additional the normal xi Part m STATISTICAL INFERENCE Chapter 8 Statistical EstiJn.ation Methods of point estimation 8.11 The method of moments 8.12 The method of maximum likelihood 8.2 Some properties of estimators 8.3 Interval estimation Additional Chapter 9 Tests of Statistical Hypotheses Statistical and non-statistical hypotheses 9.11 Simple and composite hypotheses 9.2 Type I and type II errors 9.21 The critical region

11 XU CONTENTS The power of a test The best test Some practical tests Additional 297 Chapter 10 Some Non-Parametric Tests 10.1 A goodness of fit problem 10.2 Contingency tables 10.3 Kolmogorov-Smirnov statistic I 0.4 Some distribution-free tests The sign test The rank test The run test Kruskal-Wallis H-test Additional Chapter 11 Statistical Regression The scatter diagram The method of least squares Curve fitting Regression A test for regression Additional Chapter 12 Analysis of Variance 12.1 One-way classification 12.2 Estimation of the parameters 12.3 Two-way classification

12 CONTENTS xiii Additional BibHography Answers Answers to ' Additional ' Statistical Tables I. The Binomial coefficients 2. Cummulative Binomial probabilities 3. Cummulative Poisson probabilities 4. Normal probability table 5. Student-t table 6. The chi-square table 7. The 5% point off-distribution 8. The F-Distribution 9. Random numbers Index

13 List of Symbols E--element of -not an element of a 1,.., an-set ofnumbers a 1,,an >-null set ::s;; -less than or equal to c-contains <t -not contained in.e-sum 'IT-product x-mean G.M.-geometric mean H.M.-harmonic mean S.D.-standard deviation R.M.S.D.-Root mean square deviation M.D.-mean deviation m',-r-th moment about the origin m,-r-th central moment M',-r-th absolute moment u-union n -intersection A-complement of A (n),-permutations (:)-combinations ~-approximately equal to P(A)-probability of the event A P(B \A)-probability of B given A H(Pt,...,pn)--entropy \.!-absolute value

14 LIST OF SYMBOLS XV s.v.-stochastic variable E(. )-mathematical expectation ~(r)-factorial moments var(.) -variance M(t)-moment generating function tp(t)-characteristic function y 2 -kurtosis cov(.) -covariance p-correlation coefficient L-likelihood function,\-likelihood ratio