Introduction to probability

No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or informa­ tion storage and retrieval) without permission in writing from the publisher. Preface Probability is common sense reduced to calculation Laplace This book is an outgrowth of our involvement in teaching an introductory prob­ ability course ("Probabilistic Systems Analysis'�) at the Massachusetts Institute of Technology. The course is attended by a large number of students with diverse back­ grounds, and a broad range of interests. They span the entire spectrum from freshmen to beginning graduate students, and from the engineering school to the school of management. Accordingly, we have tried to strike a balance between simplicity in exposition and sophistication in analytical reasoning. Our key aim has been to develop the ability to construct and analyze probabilistic models in a manner that combines intuitive understanding and mathematical precision. In this spirit, some of the more mathematically rigorous analysis has been just sketched or intuitively explained in the text. so that complex proofs do not stand in the way of an otherwise simple exposition. At the same time, some of this analysis is developed (at the level of advanced calculus) in theoretical prob­ lems, that are included at the end of the corresponding chapter. FUrthermore, some of the subtler mathematical issues are hinted at in footnotes addressed to the more attentive reader. The book covers the fundamentals of probability theory (probabilistic mod­ els, discrete and continuous random variables, multiple random variables, and limit theorems), which are typically part of a first course on the subject. It also contains, in Chapters 4-6 a number of more advanced topics, from which an instructor can choose to match the goals of a particular course. In particular, in Chapter 4, we develop transforms, a more advanced view of conditioning, sums of random variables, least squares estimation, and the bivariate normal distribu-v vi Preface tion. Furthermore, in Chapters 5 and 6, we provide a fairly detailed introduction to Bernoulli, Poisson, and Markov processes. Our M.LT. course covers all seven chapters in a single semester, with the ex­ ception of the material on the bivariate normal (Section 4.7), and on continuous­ time Markov chains (Section 6.5). However, in an alternative course, the material on stochastic processes could be omitted, thereby allowing additional emphasis on foundational material, or coverage of other topics of the instructor's choice. Our …