Problems and snapshots from the world of probability

Gunnar Blom, Lars Holst, Dennis Sandell

This book comprises a collection of 125 problems and snapshots from discrete probability. The problems are selected on the basis of their historical interest whereas the snapshots provide quick overviews of topics in probability such as Markov chains, Poisson processes, random walks, patterns in random sequences, cover times, and embedding procedures. This book will appeal to all those who enjoy problems with probabilistic flavor. The authors presuppose a basic exposure to discrete mathematics and elementary probability. Students will find this a stimulating companion to their courses in probability. More advanced researchers will appreciate the original style of the problems, some of which may even inspire new areas for research.

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[目次]

  • 1. Welcoming problems.- 1.1 The friendly illiterate.- 1.2 Tourist with a short memory.- 1.3 The car and the goats.- 1.4 Patterns I.- 1.5 Classical random walk I.- 1.6 Number of walks until no shoes.- 1.7 Banach's match box problem.- 1.8 The generous king.- 2. Basic probability theory I.- 2.1 Remarkable conditional probabilities.- 2.2 Exchangeability I.- 2.3 Exchangeability II.- 2.4 Combinations of events I.- 2.5 Problems concerning random numbers.- 2.6 Zero-one random variables I.- 3. Basic probability theory II.- 3.1 A trick for determining expectations.- 3.2 Probability generating functions.- 3.3 People at the corners of a triangle.- 3.4 Factorial generating functions.- 3.5 Zero-one random variables II.- 3.6 Combinations of events II.- 4. Topics from early days I.- 4.1 Cardano - a pioneer.- 4.2 Birth of probability.- 4.3 The division problem.- 4.4 Huygens's second problem.- 4.5 Huygens's fifth problem.- 4.6 Points when throwing several dice.- 4.7 Bernoulli and the game of tennis.- 5. Topics from early days II.- 5.1 History of some common distributions.- 5.2 Waldegrave's problem I.- 5.3 Petersburg paradox.- 5.4 Rencontre I.- 5.5 Occupancy I.- 5.6 Stirling numbers of the second kind.- 5.7 Bayes's theorem and Law of Succession.- 5.8 Menage I.- 6. Random permutations.- 6.1 Runs I.- 6.2 Cycles in permutations.- 6.3 Stirling numbers of the first kind.- 6.4 Ascents in permutations.- 6.5 Eulerian numbers.- 6.6 Exceedances in permutations.- 6.7 Price fluctuations.- 6.8 Oscillations I.- 6.9 Oscillations II.- 7. Miscellaneous I.- 7.1 Birthdays.- 7.2 Poker.- 7.3 Negative binomial.- 7.4 Negative hypergeometric I.- 7.5 Coupon collecting I.- 7.6 Coupon collecting II.- 7.7 Menage II.- 7.8 Rencontre II.- 8. Poisson approximation.- 8.1 Similar pairs and triplets.- 8.2 A Lotto problem.- 8.3 Variation distance.- 8.4 Poisson-binomial.- 8.5 Rencontre III.- 8.6 Menage III.- 8.7 Occupancy II.- 9. Miscellaneous II.- 9.1 Birthdays and similar triplets.- 9.2 Comparison of random numbers.- 9.3 Grouping by random division.- 9.4 Records I.- 9.5 Records II.- 9.6 A modification of blackjack.- 10. Random walks.- 10.1 Introduction.- 10.2 Classical random walk II.- 10.3 One absorbing barrier.- 10.4 The irresolute spider.- 10.5 Stars I.- 10.6 Closed stopping region.- 10.7 The reflection principle.- 10.8 Ballot problem.- 10.9 Range of a random walk.- 11. Urn models.- 11.1 Randomly filled urn.- 11.2 Polya's model I.- 11.3 Polya's model II.- 11.4 Polya's model III.- 11.5 Ehrenfest's model I.- 11.6 Ehrenfest's game.- 11.7 A pill problem.- 12. Cover times.- 12.1 Introduction.- 12.2 Complete graph.- 12.3 Linear finite graph.- 12.4 Polygon.- 12.5 A false conjecture.- 12.6 Stars II.- 12.7 Inequality for cover times.- 13. Markov chains.- 13.1 Review I.- 13.2 Review II.- 13.3 Random walk: two reflecting barriers.- 13.4 Ehrenfest's model II.- 13.5 Doubly stochastic transition matrix.- 13.6 Card shuffling.- 13.7 Transition times for Markov chains.- 13.8 Reversible Markov chains.- 13.9 Markov chains with homesickness.- 14. Patterns.- 14.1 Runs II.- 14.2 Patterns II.- 14.3 Patterns III.- 14.4 A game for pirates.- 14.5 Penney's game.- 14.6 Waldegrave's problem II.- 14.7 How many patterns?.- 15. Embedding procedures.- 15.1 Drawings with replacement.- 15.2 Repetition of colours.- 15.3 Birthdays revisited.- 15.4 Coupon collecting III.- 15.5 Drawings without replacement.- 15.6 Socks in the laundry.- 15.7 Negative hypergeometric II.- 15.8 The first-to-r game I.- 16. Special topics.- 16.1 Exchangeability III.- 16.2 Martingales.- 16.3 Wald's equation.- 16.4 Birth control.- 16.5 The r-heads-in-advance game.- 16.6 Patterns IV.- 16.7 Random permutation of 1's and (?1)'s.- 17. Farewell problems.- 17.1 The first-to-r game II.- 17.2 Random walk on a chessboard.- 17.3 Game with disaster.- 17.4 A rendezvous problem.- 17.5 Modified coin-tossing.- 17.6 Palindromes.- References.

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この本の情報

書名 Problems and snapshots from the world of probability
著作者等 Blom, Gunnar
Holst, Lars
Sandell, Dennis
出版元 Springer-Verlag
刊行年月 c1994
ページ数 xii, 240 p.
大きさ 24 cm
ISBN 3540941614
0387941614
NCID BA2175849X
※クリックでCiNii Booksを表示
言語 英語
出版国 アメリカ合衆国
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