Intermediate Probability: A Computational Approach
Intermediate Probability: A Computational Approach
Intermediate Probability is the natural extension of the author's Fundamental Probability. It details several highly important topics, from standard ones such as order statistics, multivariate normal, and convergence concepts, to more advanced ones which are usually not addressed at this mathematical level, or have never previously appeared in textbook form. The author adopts a computational approach throughout, allowing the reader to directly implement the methods, thus greatly enhancing the learning experience and clearly illustrating the applicability, strengths, and weaknesses of the theory.The book:
Places great emphasis on the numeric computation of convolutions of random variables, via numeric integration, inversion theorems, fast Fourier transforms, saddlepoint approximations, and simulation. Provides introductory material to required mathematical topics such as complex numbers, Laplace and Fourier transforms, matrix algebra, confluent hypergeometric functions, digamma functions, and Bessel functions. Presents full derivation and numerous computational methods of the stable Paretian and the singly and doubly non-central distributions. A whole chapter is dedicated to mean-variance mixtures, NIG, GIG, generalized hyperbolic and numerous related distributions. A whole chapter is dedicated to nesting, generalizing, and asymmetric extensions of popular distributions, as have become popular in empirical finance and other applications. Provides all essential programming code in Matlab and R. The user-friendly style of writing and attention to detail means that self-study is easily possible, making the book ideal for senior undergraduate and graduate students of mathematics, statistics, econometrics, finance, insurance, and computer science, as well as researchers and professional statisticians working in these fields.
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Title of
ebook: Intermediate Probability: A Computational ApproachISBN: 9780470035054Publisher:
John Wiley & Sons, Ltd.
Internet download file size: 4980 kbPages: 430Released online for download : 09-27-2007Author of eBook: Paolella, Marc
Intermediate Probability
A Computational Approach
Chapter One
Generating functions
The shortest path between two truths in the real domain passes through the
complex domain. (Jacques Hadamard)
There are various integrals of the form
[[integral].sup.[infinity].sub.-[infinity]] g(t, x) d[F.sub.X](x) = E[g(t,X)] (1.1)
which are often of great value for studying r.v.s. For example, taking g(n, x) = [x.sup.n] and
g(n, x) = [[absolute value of x].sup.n], for n [member of] [??], give the algebraic and absolute moments, respectively, while g(n, x) = [x.sub.[n]] = x(x - 1) --- (x - n + 1) yields the factorial moments of X, which are of use for lattice r.v.s. Also important (if not essential) for
working with lattice distri ... read full excerpt from Intermediate Probability: A Computational Approach ebook
