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# Volume 28 - Issue 1 - June 2003

### Mathematical modelling of homeless populations

**J. G. Byatt-Smith, A. A. Lacey, D. F. Parker, D. Simpson, W. R. Smith and J. A. D. Wattis**

Abstract

A simple model is proposed for the changes in numbers of homeless households, of those of people housed in the private sector and of those resident in council housing. The model, which applies to a single local authority, is analysed to see how changing priorities can affect waiting times and the size of the waiting list for council accommodation. The analysis shows that, if the number of homeless is reasonably small, then altering the priority given to them makes little difference to the waiting times for other members of the population. However, lowering the priority given to rehousing the homeless significantly increases the time which they have to wait. Time scales which appear in the model indicate that the determination of steady states will not always suffice to predict the sizes of waiting lists over times of practical interest. A possible sensitive dependence on the amount of housing stock is also found.

### Orthogonal trajectories in polar form

**Behzad Salimi**

Abstract

Simple procedures are introduced to calculate the families of orthogonal trajectories in polar coordinates. Basic concepts from rudimentary calculus, vector calculus, and the calculus of an analytic function of a complex variable are used to derive either a first-order ordinary differential equation to solve or a quadrature to evaluate. An example is provided to demonstrate the application of the formulae derived.

### Sets of binary random variables with a prescribed independence/dependence structure

**Jordan Stoyanov**

Abstract

This paper studies the independence/dependence (i/d) properties of a set of n ¸ 2 binary random variables. The analysis of this set is based on its i/d-structure (i2,…, ik ,…, in), where ik is the number of true product relations at level k, k = 2, …, n (level k involves all combinations of size k). The following problem is considered: `how can we construct a probability space and define a set of n binary random variables with a prescribed i/d-structure?' The solution is provided and explicitly nonstandard examples described. Related topics are also discussed.

### Perceived highway speed

**Randall Swift, Jennifer Switkes and Stephen Wirkus**

Abstract

We consider a driver's perception of the average speed on a highway based on a continuous probability distribution of car speeds. We examine the effect that the distribution of car speeds has on a given driver's perception of the mean, median, and mode. Some surprising as well as expected results are obtained.

### On the history of Bayes's theorem

**Oscar Sheynin**

Abstract

This paper reconsiders the history of Bayes's theorem, and analyzes several possibilities with regard to its authorship. The conclusion reached is that Bayes was the author of the results in the memoir on `the doctrine of chances' presented to the Royal Society of London by his friend Richard Price in 1763–64.

### Real-world resolution of the two-envelope problem with a heavy-tailed distribution

**Nelson M. Blachman**

Abstract

The `exchange paradox' or `two-envelope problem' is reformulated here with statistically independent, identically distributed amounts of money in the two envelopes. The distribution is assumed to be known, and you may see the amount of money, $X, in the envelope that you are given before deciding whether to exchange it for the other one. Ordinarily, the usual Bayesian expected-outcome criterion is appropriate, and it leads to trading envelopes if $X is less than the mean of the distribution. Simulation of the case of a particular geometric distribution suggests, however, that a modification of the criterion may be desirable when the distribution is heavy tailed, namely a maximum expected outcome under the given distribution when confined to those values that are likely to arise during a suitably large but finite number of trials. The optimal strategy would then be to exchange envelopes if and only if $X is less than the mean of the truncated distribution.

### A log-Laplace growth rate model

**Tomasz J. Kozubowski and Krzysztof Podgórski**

Abstract

Log-Laplace distributions arise as exponential functions of skew Laplace laws, and have power-tail behavior at zero and infinity. We review the basic properties of log-Laplace laws and derive their new stability property; this may explain the increasing popularity of log-Laplace laws in modeling growth rates. A numerical example where log-Laplace distributions are fitted to currency exchange rates illustrates this new property.

### On the asymptotic normality of the posterior distribution: II

**W. F. Scott**

Abstract

Let a random sample of size n be drawn from a variable whose distribution (which may be discrete, continuous or of mixed type) depends on a vector of parameters, µ. We provide some theorems on the asymptotic posterior distribution of µ. These results are frequently used in practice to find confidence limits for unknown parameters when the number of observations is large. The present paper expands the results of Walker (1969) and Scott (2001), and gives some actuarial and other applications.