The Theory of Probability: Explorations and Applications (Instructor Solution Manual, Selected Solutions)

Description

Several generations of students and teaching assistants worked out problems and contributed to the selected solutions assembled here. I should especially acknowledge Wei Bi, Shao Chieh Fang, Gaurav Kasbekar, Jonathan Nukpezah, Alireza Tahbaz Salehi, Shahin Shahrampour, Evangelos Vergetis, and Zhengwei Wu. I must confess immediately that not all the solutions have seen the same level of scrutiny; and, while I have expended some effort to bring some uniformity of style and presentation to the material, it may occasionally lack some polish. It struck me, however, that a reader desirous of an early view of the results may be willing to forgive the deficiencies in the current crude assemblage and, accordingly, that it may be preferable to provide a rough and ready collection of solutions now, however unpolished, unvetted, and incomplete, and not wait till I had had the time to proof-read them carefully and present them in a uniform format and style while filling in the gaps. References to ToP are, of course, to The Theory of Probability; references to Equations, Examples, Theorems, Sections, Chapters, and so on point to the main text; the referencing conventions are as in ToP. As a navigational aid, page headings are labelled by chapter and problem; the detailed index that I have provided may also be of use. There are inevitably errors in the text lying in wait to be discovered. I can only hope that these are of the obvious kind that do not cause great consternation and apologise for these in advance. I would certainly very much appreciate receiving word of errors and ambiguities both here and in the main text of ToP. January 1, 2015: The task of compiling solutions proceeds slowly. The current version of the palimpsest provides solutions for more than 75% of the problems: of the 560 problems assigned for solution in ToP, 428 now have solutions spelled out in detail. In many instances I have provided two or three different approaches to solution to illustrate different perspectives; and the solutions frequently flesh out themes and generalisations suggested by the patterns of attack. Table 1 on the following page identifies Contents Preface vii A ELEMENTS I Probability Spaces 3 II Conditional Probability 25 III A First Look at Independence 51 IV Probability Sieves 65 V Numbers Play a Game of Chance 87 VI The Normal Law 105 VII Probabilities on the Real Line 121 VIII The Bernoulli Schema 143 IX The Essence of Randomness 169 X The Coda of the Normal 211 B FOUNDATIONS XI Distribution Functions and Measure 247 XII Random Variables 257 XIII Great Expectations 279 XIV Variations on a Theme of Integration 293 XV Laplace Transforms 331 XVI The Law of Large Numbers 357 XVII From Inequalities to Concentration 377 XVIII Poisson Approximation 385 XIX Convergence in Law, Selection Theorems 397 XX Normal Approximation 407 Index 419