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Volume 23 - Issue 2 - December 1998
N. H. Bingham
This article is an edited version of the text of the author’s Inaugural Lecture as Professor of Statistics at Birkbeck College, University of London, delivered there on 11 June 1997. After a brief outline of the development of Brownian motion as a model of random fluctuations, a survey is given of a range of topics in which it has proved useful.
A. I. Dale
De Méré's paradox is commonly found in statistical texts. Few books however give any information about its proposer, and the details in those that do are sometimes contradictory. Moreover, the problem itself seems to be a simple exercise in elementary probability. Thus we seem to be faced with two questions: firstly, who was this de Méré, and secondly, what makes the problem a paradox?
G. S. Georgiev
Some probability problems arising in the game of Bingo are considered. Probabilities for striking a Line and completing a Bingo are calculated for one participating card. The expectations of the number of balls drawn before striking a Line or winning a Bingo are calculated.
The aim of this note is to show that several old and new results in combinatorics and probability theory can be proved very simply by using a specific property of cyclic permutations.
Lennart Bondesson and Lars Holst
Consider sums of independent random variables all having the same probability distribution function F. Convergence of such suitably normalized sums to stable limits, including normal ones, is studied using Laplace transforms instead of characteristic functions. The basic idea is to split every summand into its positive and negative parts, and then use a bivariate continuity theorem for Laplace transforms. Results for slowly varying functions and Tauberian theorems are used to connect the behaviour of the tails of F with the behaviour of the Laplace transforms at the origin.
Matt Davison and Christopher Essex
An alternative definition of a fractional differential operator corresponding to the Riemann--Liouville fractional integral is introduced. The paper demonstrates that from a naturally-arising selection of possible definitions, the alternative version is the only one suitable for normal initial value problems in the context of fractional calculus, while the standard definition is preferable for certain integral equations.