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# Volume 27 - Issue 2 - December 2002

### Obituary: Evgeny Evgenievich Slutsky

**A. N. Kolmogorov (translated by O. Sheynin)**

### The optimal extraction of a privately owned renewable resource

**David Chappell and Kevin Dowd**

Abstract

This paper provides an optimal control analysis of the extraction of a privately owned renewable resource. Our treatment integrates the extraction decision with the consumption decision, provides a complete and explicit analysis of the underlying economic problem, as well as a full analysis of the impact of parametric changes on the steady state.

### The Collins case: a `beads in urns' model

**T. Rolf Turner**

Abstract

The Collins case is an example of the gross misapplication of probability in a legal or forensic context. As such, it has been discussed by a variety of authors who lay bare the fallaciousness of the pseudo-probabilistic reasoning which was applied in the case. The reasoning which ought to be applied in such cases has, however, not been supplied in a clear and explicit manner. In this note the Collins case is described, giving references to some of the discussions of it which have been published. It is then shown how the `beads in urns' paradigm leads us quite simply to the appropriate probability model. The calculations necessary to produce the answer to the relevant probability question turn out to be completely elementary. The same analysis applies directly to the often controversial issue of assessing DNA evidence.

### Asymptotics for the maximum in the coupon collector's problem

**Michael Scheutzow**

Abstract

We show that when N dollar bills are randomly distributed among N people, the number of dollars which the luckiest person receives grows like (log N) / log log N as N ! 1. This answers a question posed by C. Pommerenke, which may be reformulated as the following problem: suppose we collect N coupons from a series of N different types; what is the maximal number of coupons of the same type? In addition, we show that the distribution of the amount the luckiest person receives is concentrated on at most two values as N ! 1. Our main technique is the well-known Poissonization method.

### Hermite polynomials and Brownian motion

**Tom Carroll**

Abstract

We present an elementary approach to the problem of determining which polynomials give rise to martingales when applied to a Brownian motion. The widely known solution involves the Hermite polynomials.

### Applying the Bernoulli equation to solve a Klein–Gordon-type equation

**Sirendaoreji and S. Jiong**

Abstract

Using solutions of a Bernoulli equation instead of the tanh(kz) in the tanh-function method, we find some more general solutions to a Klein–Gordon-type equation. For this equation, we show that there are many choices for the balancing number m and the power n of the nonlinear term in the Bernoulli equation. These lead us not only to recover the previously known solutions, but also to derive other new square root tanh-type solutions.

### Analysis of the busy period of the chemical queue: a series approach

**A. M. K. Tarabia and A. H. El-Baz**

Abstract

Conolly, Parthasarathy and Dharmaraja (1997) have obtained an expression for the joint distribution of the duration and number of customers served during a busy period for the so-called `chemical' queue. This is an exponentially driven system in which the mean arrival and service rates depend on the parity of the number of customers present. The present paper gives an alternative analysis using a series representation based on a method of Sharma and Tarabia (2000). This has the advantage of avoiding complex arithmetic, generating function manipulations and Bessel functions. We show that our formulae offer significant computational advantages.

### Some parametric chain models

**G. Jones and C. D. Lai**

Abstract

A chain model is a sequence of random variables {Xn : n = 1, 2,…} such that the conditional distribution of Xn | Xn-1 is a parametric model, with the parameters being functions of X n-1. In this paper, we point out some common, as well as some less well-known examples of chain models. We use the general framework to suggest two new models: the chain negative binomial and the chain Pareto. We investigate various properties, in particular the unconditional distributions of the random variables {Xn} and the limiting distribution of the Markov chain. Some possible applications of these models are outlined.