Continuous Distributions 45 mins quiz, total 4 questions.
Include: Probability densities, Exponential and gamma distributions, change of Variable, cumulative distribution Functions
978-0-387-94594-1_Book_PrintPDF.pdf Springer New York Berlin Heidelberg Barcelona Budapest Hong Kong London Milan Paris Tokyo Springer Texts in Statistics Advisors: Stephen Fienberg Ingram Olkin Springer Texts in Statistics Alfred: Elements of Statistics for the Life and Social Sciences Berger: An Introduction to Probability and Stochastic Processes Blom: Probability and Statistics: Theory and Applications Brockwell and Davis: An Introduction to Times Series and Forecasting Chow and Teicher: Probability Theory: Independence, Interchangeability, Martingales, Second Edition Christensen: Plane Answers to Complex Questions: The Theory of Linear Models, Second Edition Christensen: Linear Models for Multivariate, Time Series, and Spatial Data Christensen: Log-Linear Models Creighton: A First Course in Probability Models and Statistical Inference du Toit, Steyn and Stump!' 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Probability / Jim Pitman. p. cm. -- (Springer texts in statistics) Includes bibliographical references and index. ISBN -13: 978-0-387 -94594-1 1. Probabilities. 1. Title. II. Series. QA273.P493 1993 519.2--dc20 92-39051 © 1993 Springer-Verlag New York, Inc. Softcover reprint of the ha.rdcover 1st edition 1993 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analy- sis. Use in connection with any form of information storage and retrieval, electronic adapta- tion, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Production managed by Karen Phillips; manufacturing supervised by Vincent Scelta. Photocomposed pages prepared using the author's LATEX file. 9 I) 7 6 (Corrected six1:h printing, 19(7) ISBN -13: 978-0-387-94594-1 e- ISBN -13:978-1-4612-4374-8 DOl: 10.1007/978-1-4612-4374-8 Preface Preface to the Instructor This is a text for a one-quarter or one-semester course in probability, aimed at stu- dents who have done a year of calculus. The book is organized so a student can learn the fundamental ideas of probability from the first three chapters without reliance on calculus. Later chapters develop these ideas further using calculus tools. The book contains more than the usual number of examples worked out in detail. It is not possible to go through all these examples in class. Rather, I suggest that you deal quickly with the main points of theory, then spend class time on problems from the exercises, or your own favorite problems. The most valuable thing for students to learn from a course like this is how to pick up a probability problem in a new setting and relate it to the standard body of theory. The more they see this happen in class, and the more they do it themselves in exercises, the better. The style of the text is deliberately informal. My experience is that students learn more from intuitive explanations, diagrams, and examples than they do from theo- rems and proofs. So the emphasis is on problem solving rather than theory. Order of Topics. The basic rules of probability all appear in Chapter 1. Intuition for probabilities is developed using Venn and tree diagrams. Only finite additivity of probability is treated in this chapter. Discussion of countable additivity is postponed to Section 3.4. Emphasis in Chapter 1 is on the concept of a probability distribution and elementary applications of the addition and multiplication rules. Combinatorics appear via study of the binomial and hypergeometric distributions in Chapter 2. The vi Preface concepts of mean and standard deviation appear in a preliminary form in this chapter, motivated by the normal approximation, without the notation of random variables. These concepts are then developed for discrete random variables in Chapter 3. The main object of the first three chapters is to get to the circle of ideas around the normal approximation for sums of independent random variables. This is achieved by Section 3.3. Sections 3.4 and 3.5 deal with the standard distributions on the non- negative integers. Conditional distributions and expectations, covariance and corre- lation for discrete distributions are postponed to Chapter 6, nearby treatment of the same concepts for continuous distributions. The discrete theory could be done right after Chapter 3, but it seems best to get as quickly as possible to continuous things. Chapters 4 and 5 treat continuous distributions assuming a calculus background. The main emphasis here is on how to do probability calculations rather than rigorous development of the theory. In particular, differential calculations are used freely from Section 4.1 on, with only occasional discussion of the limits involved. Optional Sections. These are more demanding mathematically than the main stream of ideas. Terminology. Notation and terms are standard, except that outcome space is used throughout instead of sample space. Elements of an outcome space are called pos- sible outcomes. Pace. The earlier chapters are easier than later ones. It is important to get quickly through Chapters 1 and 2 (no more than three weeks). Chapter 3 is more substantial and deserves more time. The end of Chapter 3 is the natural time for a midterm examination. This can be as early as the sixth week. Chapters 4, 5, and 6 take time, much of it spent teaching calculus. Preface to the Student Prerequisites. This book assumes some background of mathematics, in particular, calculus. A summary of what is taken for granted can be found in Appendices I to III. Look at these to see if you need to review this material, or perhaps take another mathematics course before this one. How to read this book. To get most benefit from the text, work one section at a time. Start reading each section by skimming lightly over it. Pick out the main ideas, usually boxed, and see how some of the examples involve these ideas. Then you may already be able to do some of the first exercises at the end of the section, which you should try as soon as possible. Expect to go back and forth between the exercises and the section several times before mastering the material. Exercises. Except perhaps for the first few exercises in a section, do not expect to be able to plug into a formula or follow exactly the same steps as an example in the text. Rather, expect some variation on the main theme, perhaps a combination with ideas of a previous section, a rearrangement of the formula, or a new setting of the same principles. Through working problems you gain an active understanding of Preface vii the concepts. If you find a problem difficult, or can't see how to start, keep in mind that it will always be related to material of the section. Try re-reading the section with the problem in mind. Look for some similarity or connection to get started. Can you express the problem in a different way? Can you identify relevant variables? Could you draw a diagram? Could you solve a simpler problem? Could you break up the problem into simpler parts? Most of the problems will yield to this sort of approach once you have understood the basic ideas of the section. For more on problem-solving techniques, see the book How to Solve It by G. Polya (Princeton University Press). Solutions. Brief solutions to most odd numbered exercises appear at the end of the book. Chapter Summaries. These are at the end of every chapter. Review Exercises. These come after the summaries at the end of every chapter. Try these exercises when reviewing for an examination. Many of these exercises combine material from previous chapters. Distribution Summaries. These set out the properties of the most important distri- butions. Familiarity with these properties reduces the amount of calculation required in many exercises. Examinations. Some midterm and final examinations from courses taught from this text are provided, with solutions a few pages later. Acknowledgments Thanks to many students and instructors who have read preliminary versions of this book and provided valuable feedback. In particular, David Aldous, Peter Bickel, Ed Chow, Steve Evans, Roman Fresnedo, David Freedman, Alberto Gandolfi, Hank Ib- ser, Barney Krebs, Bret Larget, Russ Lyons, Lucien Le Cam, Maryse Loranger, Deborah Nolan, David Pollard, Roger Purves, Joe Romano, Tom Salisbury, David Siegmund, Anne Sheehy, Philip Stark, and Ruth Williams