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

✍ Scribed by Santosh S. Venkatesh

• Publisher: Cambridge University Press
• Year: 2015
• Tongue: English
• Leaves: 434
• Category: Library

✦ Synopsis

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

✦ Table of Contents

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