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(0-486-46898-4) PARTIAL DIFFERENTIAL EQUATIONS: SOURCES AND SOLUTIONS, Arthur David Snider. (0- 486-45340-5) INTRODUCTION TO BIOSTATISTICS: SECOND EDITION, Robert R. Sokal and F. James Rohlf. (0-486-46961-1) MATHEMATICAL PROGRAMMING, Steven Vajda. (0-486-47213-2) THE LOGIC OF CHANCE, John Venn. (0-486-45055-4) THE CONCEPT OF A RIEMANN SURFACE, Hermann Weyl. (0-486-47004-0) INTRODUCTION TO PROJECTIVE GEOMETRY, C. R. Wylie, Jr. (0-486-46895-X) FOUNDATIONS OF GEOMETRY, C. R. Wylie, Jr. (0-486-47214-0) See every Dover book in print at www.doverpublications.com Copyright © 1967. 1979 by M. G. Bulmer. All rights reserved. This Dover edition, first published in 1979, is an unabridged and corrected republication of the second (1967) edition of the work first published by Oliver and Boyd, Edinburgh, in 1965. Library of Congress Catalog Card Number: 78-72991 International Standard Book Number 9780486135205 Manufactured in the United States by Courier Corporation 63760315 www.doverpublications.com Table of Contents DOVER BOOKS ON MATHEMATICS Title Page Copyright Page PREFACE TO FIRST EDITION PREFACE TO SECOND EDITION CHAPTER 1 - THE TWO CONCEPTS OF PROBABILITY CHAPTER 2 - THE TWO LAWS OF PROBABILITY CHAPTER 3 - RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS CHAPTER 4 - DESCRIPTIVE PROPERTIES OF DISTRIBUTIONS CHAPTER 5 - EXPECTED VALUES CHAPTER 6 - THE BINOMIAL, POISSON AND EXPONENTIAL DISTRIBUTIONS CHAPTER 7 - THE NORMAL DISTRIBUTION CHAPTER 8 - THE χ2, t AND F DISTRIBUTIONS CHAPTER 9 - TESTS OF SIGNIFICANCE CHAPTER 10 - STATISTICAL INFERENCE CHAPTER 11 - POINT ESTIMATION CHAPTER 12 - REGRESSION AND CORRELATION REFERENCES INDEX PREFACE TO FIRST EDITION The aim of this book is to describe as simply as possible the fundamental principles and concepts of statistics. I hope that it may be useful both to the mathematician who requires an elementary introduction to the subject and to the scientist who wishes to understand the theory underlying the statistical methods which he uses in his own work. Most of the examples are of a biological nature since my own experience and interests lie in this field. I am grateful to the Biometrika Trustees for permission to reproduce Fig. 15 and the tables at the end of the book, and to Professor E. W. Sinnott and the National Academy of Sciences for permission to reproduce Fig. 23. I wish to thank Mr. J. F. Scott for reading and criticising the manuscript and Mrs. K. M. Earnshaw for assistance with the typing. M. G. B. Oxford, 1965 PREFACE TO SECOND EDITION In this edition I have added a number of exercises and problems at the end of each chapter. The exercises are straightforward and are mostly numerical, whereas the problems are of a more theoretical nature and many of them extend and expand the exposition of the text. The exercises should not present much difficulty but the problems demand more mathematical maturity and are primarily intended for the mathematician. However, I recommend mathematicians to do the exercises first and to reserve the problems until a second reading since it is important for them to acquire an intuitive feeling for the subject before they study it in detail. Some of the exercises and problems are taken from the Preliminary Examination in Psychology, Philosophy and Physiology and from the Certificate and Diploma in Statistics of Oxford University and I am grateful to the Clarendon Press for permission to print them. I have also taken the opportunity to rewrite the sections on correlation in Chapters 5 and 12, and to correct a number of misprints, for whose detection I am indebted to Professor P. Armitage, Miss Susan Jennings, and a number of other readers. M. G. B. Oxford, 1966 CHAPTER 1 THE TWO CONCEPTS OF PROBABILITY “But ‘glory’ doesn’t mean ‘a nice knock-down argument ’,” Alice objected. “When I use a word,” Humpty Dumpty said, in rather a scornful tone, “it means just what I choose it to mean—neither more nor less.” “The question is,” said Alice, “whether you can make words mean so many different things.” “The question is,” said Humpty Dumpty, “which is to be master—that’s all.” Lewis Carroll: Through the Looking-glass It is advisable in any subject to begin by defining the terms which are to be used. This is particularly important in probability theory, in which much confusion has been caused by failure to define the sense in which the word probability is being used. For there are two quite distinct concepts of probability, and unless they are carefully distinguished fruitless controversy can arise between people who are talking about different things without knowing it. These two concepts are: (1) the relative frequency with which an event occurs in the long run, and (2) the degree of belief which it is reasonable to place in a proposition on given evidence. The first of these I shall call statistical probability and the second inductive probability, following the terminology of Darrell Huff (1960). We will now discuss them in turn. Much of this discussion, particularly that of the logical distinction between the two concepts of probability, is based on the excellent account in the second chapter of Carnap (1950). STATISTICAL PROBABILITY The concept of statistical probability is based on the long run stability of frequency ratios. Before giving a definition I shall illustrate this idea by means of two examples, from a coin-tossing experiment and from data on the numbers of boys and girls born. Coin-tossing. No one can tell which way a penny will fall; but we expect the proportions of heads and tails after a large number of spins to be nearly equal. An experiment to demonstrate this point was performed by Kerrich while he was interned in Denmark during the last war. He tossed a coin 10,000 times and obtained altogether 5067 heads; thus at the end of the experiment the proportion of heads was •5067 and that of tails •4933. The behaviour of the proportion of heads throughout the experiment is shown in Fig. 1.