| Description |
1 online resource (xxiv, 805 pages) : illustrations |
| Contents |
880-01 Probability spaces -- Conditional probability -- A first look at independence -- Probability sieves -- Numbers play a game of chance -- The normal law -- Probabilities on the real line -- The Bernoulli schema -- The essence of randomness -- The coda of the normal -- Distribution functions and measure -- Random variables -- Great expectations -- Variations on a theme of integration -- Laplace transforms -- The law of large numbers -- From inequalities to concentration -- Poisson approximation -- Convergence in law, selection theorems -- Normal approximation -- Appendix: Sequences, functions, spaces |
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880-01/(S Machine generated contents note: A. Elements -- I. Probability Spaces -- 1. From early beginnings to a model theory -- 2. Chance experiments -- 3. sample space -- 4. Sets and operations on sets -- 5. algebra of events -- 6. probability measure -- 7. Probabilities in simple cases -- 8. Generated a-algebras, Borel sets -- 9. little point set topology -- 10. Problems -- II. Conditional Probability -- 1. Chance domains with side information -- 2. Gender biasSimpson's paradox -- 3. theorem of total probability -- 4. Le probleme des rencontres, matchings -- 5. Polya's urn scheme, spread of contagion -- 6. Ehrenfest model of diffusion -- 7. Bayes's rule for events, the MAP principle -- 8. Laplace's law of succession -- 9. Back to the future, the Copernican principle -- 10. Ambiguous communication -- 11. Problems -- III. First Look at Independence -- 1. rule of products -- 2. What price intuition-- 3. 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. sieve of Eratosthenes -- 3. On trees and a formula of Cayley -- 4. Boole's inequality, the Borel-Cantelli lemmas -- 5. Applications in Ramsey theory -- 6. Bontferroni's inequalities, Poisson approximation -- 7. Applications in random graphs, isolation -- 8. Connectivity, from feudal states to empire -- 9. Sieves, the Lovasz 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. formula of Viete -- 2. Binary digits, Rademacher functions -- 3. independence of the binary digits -- 4. link to coin tossing -- 5. binomial makes an appearance -- 6. inequality of Chebyshev -- 7. Borel discovers numbers are normal -- 8. Problems -- VI. Normal Law -- 1. One curve to rule them all -- 2. little Fourier theory I -- 3. little Fourier theory II -- 4. idea of Markov -- 5. Levy suggests a thin sandwich, de Moivre redux -- 6. local limit theorem -- 7. Large deviations -- 8. 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. Bernoulli Schema -- 1. Bernoulli trials -- 2. binomial distribution -- 3. On the efficacy of polls -- 4. simple random walk -- 5. 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. curious case of the tennis rankings -- 10. Population size, the hypergeometric distribution -- 11. Problems -- IX. Essence of Randomness -- 1. uniform density, a convolution formula -- 2. Spacings, a covering problem -- 3. Lord Rayleigh's random flights -- 4. M. Poincare joue a 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. Coda of the Normal -- 1. normal density -- 2. Squared normals, the chi-squared density -- 3. little linear algebra -- 4. multivariate normal -- 5. application in statistical estimation -- 6. Echoes from Venus -- 7. strange case of independence via mixing -- 8. continuous, nowhere differentiable function -- 9. Brownian motion, from phenomena to models -- 10. Haar system, a curious identity -- 11. bare hands construction -- 12. paths of Brownian motion are very kinky -- 13. Problems -- B. Foundations -- XI. Distribution Functions and Measure -- 1. Distribution functions -- 2. Measure and its completion -- 3. Lebesgue measure, countable sets -- 4. measure on a ring -- 5. Prom measure to outer measure, and back -- 6. Problems -- XII. Random Variables -- 1. Measurable maps -- 2. 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. 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 -- 2. Change of variable, moments, correlation -- 3. Inequalities via convexity -- 4. Lp-spaces -- 5. Iterated integrals, a cautionary example -- 6. volume of an n-dimensional ball -- 7. asymptotics of the gamma function -- 8. question from antiquity -- 9. How fast can we communicate-- 10. Convolution, symmetrisation -- 11. Labeyrie ponders the diameter of stars -- 12. Problems -- XV. Laplace Transforms -- 1. transform of a distribution -- 2. Extensions -- 3. renewal equation and process -- 4. Gaps in the Poisson process -- 5. Collective risk and the probability of ruin -- 6. queuing process -- 7. Ladder indices and a combinatorial digression -- 8. amazing properties of fluctuations -- 9. Polya walks the walk -- 10. Problems -- XVI. Law of Large Numbers -- 1. Chebyshev's inequality, reprise -- 2. Khinchin's law of large numbers -- 3. physicist draws inspiration from Monte Carlo -- 4. Triangles and cliques in random graphs -- 5. gem of Weierstrass -- 6. Some number-theoretic sums -- 7. dance of the primes -- 8. Fair games, the St |
| Summary |
From classical foundations to modern theory, this comprehensive guide to probability interweaves mathematical proofs, historical context and detailed illustrative applications |
| Bibliography |
Includes bibliographical references and index |
| Notes |
English |
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Print version record |
| Subject |
Probabilities.
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Probabilities -- History
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Probability
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probability.
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MATHEMATICS -- Probability & Statistics -- General.
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Probabilities
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| Genre/Form |
History
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| Form |
Electronic book
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| ISBN |
9781139840316 |
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1139840312 |
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9781139841504 |
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1139841505 |
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9781299409248 |
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1299409245 |
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9781139854139 |
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1139854135 |
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9781139169325 |
|
1139169327 |
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9781139842693 |
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1139842692 |
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9781139845052 |
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1139845055 |
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9781139845052 |
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1107236134 |
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9781107236134 |
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1107254612 |
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9781107254619 |
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