# 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,
J. A. D. Wattis
pp. 112 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
pp. 1318 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
pp. 1927 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, Stephen Wirkus
pp. 2836 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
pp. 3742 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
pp. 4348 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, Krzysztof Podgórski
pp. 4960 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
pp. 6166 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.