The Theory of Probability :Explorations and Applications

Publication subTitle :Explorations and Applications

Author: Santosh S. Venkatesh  

Publisher: Cambridge University Press‎

Publication year: 2012

E-ISBN: 9781139848220

P-ISBN(Paperback): 9781107024472

Subject: O211 probability (probability theory, probability theory)

Keyword: 概率论与数理统计

Language: ENG

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The Theory of Probability

Description

From classical foundations to advanced modern theory, this self-contained and comprehensive guide to probability weaves together mathematical proofs, historical context and richly detailed illustrative applications. A theorem discovery approach is used throughout, setting each proof within its historical setting and is accompanied by a consistent emphasis on elementary methods of proof. Each topic is presented in a modular framework, combining fundamental concepts with worked examples, problems and digressions which, although mathematically rigorous, require no specialised or advanced mathematical background. Augmenting this core material are over 80 richly embellished practical applications of probability theory, drawn from a broad spectrum of areas both classical and modern, each tailor-made to illustrate the magnificent scope of the formal results. Providing a solid grounding in practical probability, without sacrificing mathematical rigour or historical richness, this insightful book is a fascinating reference and essential resource, for all engineers, computer scientists and mathematicians.

Chapter

8 Laplace’s law of succession

9 Back to the future, the Copernican principle

10 Ambiguous communication

11 Problems

III A First Look at Independence

1 A rule of products

2 What price intuition?

3 An application in genetics, Hardy’s law

4 Independent trials

5 Independent families, Dynkin’s π–λ theorem

6 Problems

IV Probability Sieves

1 Inclusion and exclusion

2 The sieve of Eratosthenes

3 On trees and a formula of Cayley

THEOREM 1 Every finite tree contains at least two leaves.

THEOREM 2 Every tree on n vertices contains n − 1 edges.

4 Boole’s inequality, the Borel–Cantelli lemmas

5 Applications in Ramsey theory

6 Bonferroni’s inequalities, Poisson approximation

7 Applications in random graphs, isolation

8 Connectivity, from feudal states to empire

9 Sieves, the Lovász local lemma

10 Return to Ramsey theory

11 Latin transversals and a conjecture of Euler

12 Problems

V Numbers Play a Game of Chance

1 A formula of Viète

2 Binary digits, Rademacher functions

3 The independence of the binary digits

4 The link to coin tossing

5 The binomial makes an appearance

6 An inequality of Chebyshev

7 Borel discovers numbers are normal

8 Problems

VI The Normal Law

1 One curve to rule them all

2 A little Fourier theory I

3 A little Fourier theory II

4 An idea of Markov

5 Lévy suggests a thin sandwich, de Moivre redux

6 A local limit theorem

7 Large deviations

8 The limits of wireless cohabitation

9 When memory fails

10 Problems

VII Probabilities on the Real Line

1 Arithmetic distributions

2 Lattice distributions

3 Towards the continuum

4 Densities in one dimension

5 Densities in two and more dimensions

6 Randomisation, regression

7 How well can we estimate?

8 Galton on the heredity of height

9 Rotation, shear, and polar transformations

10 Sums and products

11 Problems

VIII The Bernoulli Schema

1 Bernoulli trials

2 The binomial distribution

3 On the efficacy of polls

4 The simple random walk

5 The arc sine laws, will a random walk return?

6 Law of small numbers, the Poisson distribution

7 Waiting time distributions

8 Run lengths, quality of dyadic approximation

9 The curious case of the tennis rankings

10 Population size, the hypergeometric distribution

11 Problems

IX The Essence of Randomness

1 The uniform density, a convolution formula

2 Spacings, a covering problem

3 Lord Rayleigh’s random flights

4 M. Poincaré joue à la roulette

5 Memoryless variables, the exponential density

6 Poisson ensembles

7 Waiting times, the Poisson process

8 Densities arising in queuing theory

9 Densities arising in fluctuation theory

10 Heavy-tailed densities, self-similarity

11 Problems

X The Coda of the Normal

1 The normal density

2 Squared normals, the chi-squared density

3 A little linear algebra

4 The multivariate normal

5 An application in statistical estimation

6 Echoes from Venus

7 The strange case of independence via mixing

8 A continuous, nowhere differentiable function

9 Brownian motion, from phenomena to models

10 The Haar system, a curious identity

11 A bare hands construction

12 The paths of Brownian motion are very kinky

13 Problems

Part B: FOUNDATIONS

XI Distribution Functions and Measure

1 Distribution functions

2 Measure and its completion

3 Lebesgue measure, countable sets

4 A measure on a ring

5 From measure to outer measure, and back

6 Problems

XII Random Variables

1 Measurable maps

2 The induced measure

3 Discrete distributions

4 Continuous distributions

5 Modes of convergence

6 Baire functions, coordinate transformations

7 Two and more dimensions

8 Independence, product measures

9 Do independent variables exist?

10 Remote events are either certain or impossible

11 Problems

XIII Great Expectations

1 Measures of central tendency

2 Simple expectations

3 Expectations unveiled

4 Approximation, monotone convergence

5 Arabesques of additivity

6 Applications of additivity

7 The expected complexity of Quicksort

8 Expectation in the limit, dominated convergence

9 Problems

XIV Variations on a Theme of Integration

1 UTILE ERIT SCRIBIT ∫PRO OMNIA

INTEGRALS WITH RESPECT TO GENERAL MEASURES

2 Change of variable, moments, correlation

3 Inequalities via convexity

4 Lp-spaces

COMPLETENESS

5 Iterated integrals, a cautionary example

6 The volume of an n-dimensional ball

7 The asymptotics of the gamma function

8 A question from antiquity

9 How fast can we communicate?

10 Convolution, symmetrisation

SYMMETRISATION

11 Labeyrie ponders the diameter of stars

ONE-DIMENSIONAL STARS

TWO-DIMENSIONAL STARS

12 Problems

XV Laplace Transforms

1 The transform of a distribution

2 Extensions

3 The renewal equation and process

4 Gaps in the Poisson process

5 Collective risk and the probability of ruin

6 The queuing process

7 Ladder indices and a combinatorial digression

8 The amazing properties of fluctuations

9 Pólya walks the walk

10 Problems

XVI The Law of Large Numbers

1 Chebyshev’s inequality, reprise

2 Khinchin’s law of large numbers

3 A physicist draws inspiration from Monte Carlo

4 Triangles and cliques in random graphs

5 A gem of Weierstrass

6 Some number-theoretic sums

7 The dance of the primes

8 Fair games, the St. Petersburg paradox

9 Kolmogorov’s law of large numbers

10 Convergence of series with random signs

11 Uniform convergence per Glivenko and Cantelli

12 What can be learnt per Vapnik and Chervonenkis

VAPNIK–CHERVONENKIS CLASSES

THE GEOMETRY OF THE SITUATION

A QUESTION OF IDENTIFICATION

13 Problems

XVII From Inequalities to Concentration

1 Exponential inequalities

2 Unreliable transcription, reliable replication

3 Concentration, the Gromov–Milman formulation

4 Talagrand views a distance

5 The power of induction

THE INDUCTION BASE

THE INDUCTION STEP

6 Sharpening, or the importance of convexity

7 The bin-packing problem

8 The longest increasing subsequence

9 Hilbert fills space with a curve

10 The problem of the travelling salesman

11 Problems

XVIII Poisson Approximation

1 A characterisation of the Poisson

2 The Stein–Chen method

3 Bounds from Stein’s equation

4 Sums of indicators

5 The local method, dependency graphs

6 Triangles and cliques in random graphs, reprise

EXTENSION: CLIQUES

7 Pervasive dependence, the method of coupling

8 Matchings, ménages, permutations

9 Spacings and mosaics

10 Problems

XIX Convergence in Law, Selection Theorems

1 Vague convergence

2 An equivalence theorem

3 Convolutional operators

4 An inversion theorem for characteristic functions

5 Vector spaces, semigroups

6 A selection theorem

7 Two by Bernstein

8 Equidistributed numbers, from Kronecker to Weyl

9 Walking around the circle

10 Problems

XX Normal Approximation

1 Identical distributions, the basic limit theorem

2 The value of a third moment

3 Stein’s method

4 Berry–Esseen revisited

5 Varying distributions, triangular arrays

6 The coupon collector

7 On the number of cycles

8 Many dimensions

9 Random walks, random flights

10 A test statistic for aberrant counts

11 A chi-squared test

12 The strange case of Sir Cyril Burt, psychologist

13 Problems

Part C: APPENDIX

XXI Sequences, Functions, Spaces

1 Sequences of real numbers

2 Continuous functions

3 Some L2 function theory

Index

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