Elements of Probability and Statistics: An Introduction to Probability with de Finetti’s Approach and to Bayesian Statistics

Springer, Jan 22, 2016 - Mathematics - 246 pages

This book provides an introduction to elementary probability and to Bayesian statistics using de Finetti's subjectivist approach. One of the features of this approach is that it does not require the introduction of sample space – a non-intrinsic concept that makes the treatment of elementary probability unnecessarily complicate – but introduces as fundamental the concept of random numbers directly related to their interpretation in applications. Events become a particular case of random numbers and probability a particular case of expectation when it is applied to events. The subjective evaluation of expectation and of conditional expectation is based on an economic choice of an acceptable bet or penalty. The properties of expectation and conditional expectation are derived by applying a coherence criterion that the evaluation has to follow. The book is suitable for all introductory courses in probability and statistics for students in Mathematics, Informatics, Engineering, and Physics.

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Contents

 1 Random Numbers 3 2 Discrete Distributions 27 3 OneDimensional Absolutely Continuous Distributions 43 4 Multidimensional Absolutely Continuous Distributions 57 5 Convergence of Distributions 73 6 Discrete Time Markov Chains 81 7 Continuous Time Markov Chains 89 8 Statistics 103
 13 Markov Chains 181 14 Statistics 197 Appendix AElements of Combinatorics 213 Appendix BRelations Between Discrete and AbsolutelyContinuous Distributions 216 Appendix CSome Discrete Distributions 217 Appendix DSome OneDimensional AbsolutelyContinuous Distributions 218 Appendix EThe Normal Distribution 219 Appendix FStirlings Formula 220

 Part II Exercises 114 9 Combinatorics 117 10 Discrete Distributions 127 11 OneDimensional Absolutely Continuous Distributions 142 12 Absolutely Continuous and Multivariate Distributions 151
 Appendix GElements of Analysis 231 Appendix HBidimensional Integrals 235 References 242 Index 243 Copyright

About the author (2016)

Massimo Campanino was born on 1952. On 1975 he got the degree on Mathematics at the University of Rome, Italy, under the supervisorship in Mathematics under the supervisorship of prof. Bruno de Finetti. Since 1990 he is full Professor of Probability at the University of Bologna, Italy. He has been visitor at I. H. E. S. of Bures-sur-Yvette (France), at the University of Princeton and at the University of California Irvine. He has performed research on random fields, dynamical systems random processes, systems with random potential and in random environment, percolation. He is the author of works in collaboration with H. Epstein and D. Ruelle where the existence of a solution of Feigenbaum’s functional equation, related to the universal behaviour of one-dimensional dynamical systems, was first proved. In works in collaboration with D. Ioffe he proved the Ornstein-Zernike behaviour for Bernoulli percolation below the critical probability and then with D. Ioffe and Y. Velenik for finite range Ising models. He has been national coordinator of the 2006 Prin project "Percolation, random fields, evolution of stochastic interacting systems" and of the 2009 Prin project "Random fields, percolation and stochastic.

Francesca Biagini was born in 1973. In Pisa she got the degree in Mathematics at the University in 1996 and attended the Scuola Normale, where she also obtained her PhD in Mathematics with specialization in Financial Mathematics in 2001. In 1999 she got a position as ricercatore at the University of Bologna. She moved then in 2005 to the University of Munich as associate professor. In 2009 she got the Chair of Financial and Insurance Mathematics at the University of Munich. She has been visiting at the University of Oslo, Evry, Toulose, Singapore, UCSB, Columbia University and Stockholm School of Economics. Her field of research concerns mainly martingale methods for financial and insurance markets and stochastic calculus for fractional Brownian motion.She is presently Member of the Council of the Bachelier Finance Society and associate editor for the journal Stochastic Analysis and Applications.