初等概率論（第4版）

前言

In this edition two new chapters, 9 and 10, on mathematical finance areadded. They are written by Dr. Farid AitSahlia, ancien dlve, who hastaught such a course and worked on the research staff of several industrialand financial institutions.The new text begins with a meticulous account of the uncommon vocab-ulary and syntax of the financial world; its manifold options and actions,with consequent expectations and variations, in the marketplace. These arethen expounded in clear, precise mathematical terms and treated by themethods of probability developed in the earlier chapters. Numerous gradedand motivated examples and exercises are supplied to illustrate the appli-cability of the fundamental concepts and techniques to concrete financialproblems. For the reader whose main interest is in finance, only a portionof the first eight chapters is a "prerequisite" for the study of the last twochapters. Further specific references may be scanned from the topics listedin the Index, then pursued in more detail.I have taken this opportunity to fill a gap in Section 8.1 and to expandAppendix 3 to include a useful proposition on martingale stopped at anoptional time. The latter notion plays a basic role in more advanced finan-cial and other disciplines. However, the level of our compendium remainselementary, as befitting the title and scheme of this textbook. We have alsoincluded some up-to-date financial episodes to enliven, for the beginners,the stratified atmosphere of "strictly business". We are indebted to RuthWilliams, who read a draft of the new chapters with valuable suggestionsfor improvement; to Bernard Bru and Marc Barbut for information on thePareto-L~vy laws originally designed for income distributions. It is hopedthat a readable summary of this renowned work may be found in the newAppendix 4.

內容概要

本書是一部介紹概率論及其應用的入門教程。其原始版本面世已經有30余年，但仍然是本科一二年級的經典概率教程。在第4版中增加了兩章講述應用和數學金融。傳承前面版本詳細、嚴謹的風格，講述了有價證券和期貨理論的基本知識。書中用最初等的方法講述了概率測度、隨機變量、分布以及期望等基本概念。離散和連續(xù)的案例都有所涉及，在講述后者的時候運用了微積分知識。配以大量的典型例子重點講述概率推理，集中介紹了組合問題、Poison過程、隨機漫步、遺傳模型和Markov鏈。每章末都附有習題及其解答。目次：集合；概率；計數；隨機變量；附錄?！　∽x者對象：數學專業(yè)的本科生以及廣大概率論愛好者。

書籍目錄

PREFACE TO THE FOURTH EDITION PROLOGUE TO INTRODUCTION TO MATHEMATICAL FINANCE 1 SET 　1.1 Sample sets 　1.2 Operations with sets 　1.3 Various relations 　1.4 Indicator 　Exercises 2 PROBABILITY 　2.1 Examples of probability 　2.2 Definition and illustrations 　2.3 Deductions from the axioms 　2.4 Independent events 　2.5 Arithmetical density 　Exercises 3 COUNTING 　3.1 Fundamental rule 　3.2 Diverse ways of sampling 　3.3 Allocation models; binomial coefficients 　3.4 How to solve it 　Exercises 4 RANDOM VARIABLES 　4.1 What is a random variable? 　4.2 How do random variables come about? 　4.3 Distribution and expectation 　4.4 Integer-valued random variables 　4.5 Random variables with densities 　4.6 General case 　Exercises 　APPENDIX 1: BOREL FIELDS AND GENERAL RANDOM VARIABLES 5 CONDITIONING AND INDEPENDENCE 　5.1 Examples of conditioning 　5.2 Basic formulas 　5.3 Sequential sampling 　5.4 P61ya's urn scheme 　5.5 Independence and relevance 　5.6 Genetical models 　Exercises 6 MEAN, VARIANCE, AND TRANSFORMS 　6.1 Basic properties of expectation 　6.2 The density case 　　6.3 Multiplication theorem; variance and covariance 　6.4 Multinomial distribution 　6.5 Generating function and the like 　Exercises 7 POISSON AND NORMAL DISTRIBUTIONS 　7.1 Models for Poisson distribution 　7.2 Poisson process 　7.3 From binomial to normal 　7.4 Normal distribution 　7.5 Central limit theorem 　7.6 Law of large numbers 　Exercises APPENDIX 2: STIRLING'S FORMULA AND DE MOIVRE-LAPLACE'S THEOREM 8 FROM RANDOM WALKS TO MARKOV CHAINS 　8.1 Problems of the wanderer or gambler 　8.2 Limiting schemes 　8.3 Transition probabilities 　8.4 Basic structure of Markov chains 　8.5 Further developments 　8.6 Steady state 　8.7 Winding up (or down?) 　Exercises APPENDIX 3: MARTINGALE 9 MEAN-VARIANCE PRICING MODEL 　9.1 An investments primer 　9.2 Asset return and risk 　9.3 Portfolio allocation 　9.4 Diversification 　9.5 Mean-variance optimization 　9.6 Asset return distributions 　9.7 Stable probability distributions 　Exercises APPENDIX 4: PARETO AND STABLE LAWS 10 OPTION PRICING THEORY 　10.1 Options basics 　10.2 Arbitrage-free pricing: 1-period model 　10.3 Arbitrage-free pricing: N-period model 　10.4 Fundamental asset pricing theorems Exercises GENERAL REFERENCES ANSWERS TO PROBLEMS VALUES OF THE STANDARD NORMAL DISTRIBUTION FUNCTION INDEX

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