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# Volume 27 - Issue 1 - June 2002

### Applying the Poisson integral formula to signal processing

**J. W. Wilder**

Abstract

The Cauchy and Poisson integral formulae are routinely covered in undergraduate courses on complex variables, along with some rather basic applications. Unfortunately, to gain an appreciation for the power of these formulae, students need a background in an applied area where boundary value problems are solved using them. Owing to the lack of such a background on the part of most students, often the only topic that is covered with reference to these formulae is the evaluation of integrals in the complex plane, which the student may have trouble connecting to real-life applications. In this paper, these theoretical tools are combined with some simple numerical work to generate surprising results involving the removal of noise from a signal, a topic which has applications in many important fields. The application of these formulae to this area requires very little previous background in signal processing, and only a basic background in complex variables.

### Euler's formula and random geometric graphs

**Jacob Benfield and Anant P. Godbole**

Abstract

We consider a probabilistic analogue of a well-known problem in combinatorial geometry. If n points are placed on a circle in `general position', then how many regions R do the crossings of the Cn2 chords generate? The answer is 1 + Cn2 + Cn4. The expected number and variance of the number of regions is computed when each chord is independently inserted with probability p. The concentration of R around its expected value, E(R), is studied, and a central limit theorem is proved.

### A new explicit solution for a chemical queue

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

Abstract

It has been demonstrated by Sharma and Bunday (1997) and later by Sharma and Tarabia (2000) that a power series technique can be successfully applied to derive certain probability distributions in queueing systems. In this paper, we further illustrate how this technique can be used to derive elegant explicit expressions for the transient state distribution of a queueing problem having `chemical' rules, i.e. in such a queue the atoms (customers) are of two alternating kinds and the clock mechanism is different according to whether the atom occupies an even or odd position on the chain. Analogous results are derived for a certain random walk on the positive and negative integers -1 < n < 1. Moreover, from this, other more commonly sought results such as the transient solution of the M/M/1/1 queue in the case λ = μ can be computed easily. A brief comparison is given between our new results and the other results mentioned in Conolly et al. (1997) in CPU time. The effectiveness of our procedure is illustrated by tables and graphs.

### A simplified log-rank test

**W. F. Scott**

Abstract

The theory of the log-rank test, which is the most popular statistical test in survival analysis, is discussed. It is contended that a martingale-based approach is over-sophisticated, and should be replaced by the concept of a `meta-analysis' of several simple rank tests of the Savage type. Similarly, Cox proportional hazards theory (which is also mathematically challenging) may be replaced by more elementary concepts.

### Solution of the inverse coupon collector's problem

**Eric Langford and Rebecca Langford**

Abstract

The classic coupon collector's problem states that: given that there are N different coupons available in boxes of a certain product, what is the probability that after buying m such boxes, exactly i different coupons will have been collected? The inverse coupon collector's problem can be stated as follows: if m boxes have esulted in i different coupons, what is the most probable value of N in the maximum likelihood sense? This problem was discussed by Dawkins (1991), who found a rough value for N by approximating the solution to a transcendental equation. We show how a solution can be obtained by solving a polynomial equation using Newton's method, which converges in this case. An application to the estimation of the size of an animal population is also provided.

### Optimal control of two competing diseases with state-dependent infection rates

**E. G. Kyriakidis and T. D. Dimitrakos**

Abstract

A two-dimensional simple stochastic epidemic process is introduced in which the infection rates depend on a power of the number of the infectives. It is assumed that one of the diseases is serious while the other is relatively harmless. Policies for introducing infection by the harmless disease or for isolating infectives with the serious disease are considered. Suitable dynamic programming algorithms are given for the determination of the policy, which minimises the expected future cost at any stage. For the corresponding deterministic model, the optimal policy is found analytically in two cases, and is compared numerically with the optimal policy for the stochastic model.

### A note on mass-action and random allocation epidemic models

**B. Barnes**

Abstract

Many mathematical models have been developed to describe epidemic processes. Deterministic models are more suitable, typically, for epidemics in large populations, while stochastic models provide an improved analysis of the epidemic process, particularly for small populations. In this note we discuss some similarities and differences between a very simple deterministic mass-action model in a homogeneously mixing population of infectives and susceptibles, and a probabilistic random allocation model, typically used to model the spread of infection from shared needles.

### Modeling The 1984–1993 American League baseball results as dependent categorical data

**H. Bayo Lawal**

Abstract

This paper gives the analysis for the 1984–1993 ten-year season results from the 14 teams in American professional baseball, comprising teams from both the Eastern and Western Divisions. The Bradley–Terry model was employed to obtain the estimated conditional probabilities of a team defeating another team for each season. The model is implemented in SAS® by using PROC GENMOD. Only the results of the 1991 season are presented using this approach. Further analysis considers the three Groeneveld scoring methods and an extension of the Bradley–Terry model incorporating a `home field' or `order effect' parameter into our model. The resulting modified Bradley–Terry model (a logit model) is implemented with PROC LOGISTIC in SAS, while the Groeneveld methods are also implemented in SAS with PROC IML. Estimated probabilities of the home teams winning are obtained for each team for each year, as well as the probability of a home team winning against an evenly matched team. The averages of these probabilities are used to rank the teams for the entire period of study. A similar approach was employed by Clarke and Norman. Our analyses indicate that the Groeneveld scoring method 3 and the modified Bradley–Terry approach rank the teams similarly. Based on this analysis, the Toronto Blue Jays were ranked first while the Cleveland Indians were ranked last overall for the entire period, 1984–1993.