PROBABILITY THEORY FOR STATISTICAL METHODS BY F. N. DAVID CAMBRIDGE AT THE UNIVERSITY PRESS 1951 To PROFESSOR JERZY NEYMAN PREFACE For some years it has been the privilege of the writer to give lectures on the Calculus of Probability to supplement courses of lectures by others on elementary statistical methods. The basis of all statistical methods is probability theory, but the teacher of mathematical statistics is concerned more with the application of fundamental probability theorems than with their proof. It is thus a convenience for both teachers and writers of textbooks on statistical methods to assume the proof of certain theorems, or at least to direct the student to a place where their proof may be found, in order that there shall be the minimum divergence from the main theme. This treatise sets out to state and prove in elementary mathe matical language those propositions and theorems of the calculus of probability which have been found useful for students of elementary statistics. It is not intended as a comprehensive treatise for the mathematics graduate the reader has been envisaged as a student with Inter. B. Sc. mathematics who wishes to teach himself statistical methods and who is desirous of supplementing his reading. With this end in view the mathe matical argument has often been set out very fully and it has always been kept as simple as possible. Such theorems as do not appear to have a direct application in statistics have not been considered and an attempt has been made at each and every stage to give practical examples. In a few cases, towards the end of the book, when it has been thought that a rigorous proof of a theorem would be beyond the scope of the readers mathematics, I have been content to state the theorem and to leave it at that. The student is to be pardoned if he obtains from the elementary algebra textbooks the idea that workers in the probability field are concerned entirely with the laying of odds, the tossing of dice or halfpennies, or the placing of persons at a dinner table. All these are undoubtedly useful in everyday life as occasion arises but they are rarely encountered in statistical practice. Hence, while I have not scrupled to use these illustrations in my turn, as viii Preface soon as possible I have tried to give examples which might be met with in any piece of statistical analysis. There is nothing new under the sun and although the elemen tary calculus of probability has extended vastly in mathematical rigour it has not advanced much in scope since the publication of Theorie des Probability by Laplace in 1812. The serious student who wishes to extend his reading beyond the range of this present book could do worse than to plod his way patiently through this monumental work. By so doing he will find how much that is thought of as modern had already been treated in a very general way by Laplace. It is a pleasure to acknowledge my indebtedness to my colleague, Mr N. L. Johnson, who read the manuscript of this book and who made many useful suggestions. I must thank my colleague, Mrs M. Merrington for help in proofreading, and the University Press, Cambridge, for the uniform excellence of their type setting. Old students of this department cannot but be aware that many of the ideas expressed here have been derived from my teacher and one-time colleague, Professor J. Neyman, now of the University of California. It has been impossible to make full acknowledgement and it is to him therefore that I would dedicate this book. Nevertheless, just as probability is, ultimately, the expression of the result of a complex of many factors on ones own mind, so this book represents the synthesis of different and often opposing ideas. In brief, while many people have given me ideas the interpretation and possible distortion of them are peculiarly mine. F. N. DEPARTMENT OP STATISTICS UNIVERSITY COLLEGE, LONDON IX CONTENTS Preface page vii Chapter I. Fundamental ideas 1 II...