Rodney Coleman

Random paths through rectangles and cubes

Abstract The sizes of particles embedded at random in an opaque material can be estimated from the lengths of paths through the particles made by a straight line probe. The length distribution of these paths is known to be related to length distributions of paths resulting from other random mechanisms. These random mechanisms for generating paths are described. When the particles are rectangles embedded in a plane surface, or cubes embedded in three-dimensional Euclidean space, the resulting path length distributions under the different randomnesses are known, but take complicated forms. Recurrence formulae are derived for calculating the moments of these distributions, and the moments are tabulated. Simple relationships between the moments under the different randomnesses are found.

Advanced multiple beam equalization radiography (AMBER) in the detection of diffuse lung disease

To evaluate the effects of scanning equalization radiography (SER) on the detection of diffuse lung disease a clinical comparison between an Advanced Multiple Beam Equalization Radiography (AMBER) unit and conventional chest radiography was performed. Even though the overall detection of focal pulmonary lesions with the AMBER unit has been shown to be significantly higher than with conventional radiography because of the improved demonstration of the costophrenic and retrocardiac regions, the utility of AMBER in the demonstration of diffuse lung disease has not been established. Twenty-one patients with diffuse lung disease (fibrosing alveolitis or sarcoidosis) and six patients with no pulmonary disease had high kVp frontal and lateral chest radiographs on both an AMBER unit and a conventional chest stand. The pooled results of five observers using Receiver Operating Characteristic (ROC) analysis indicate that there is a slight improvement but no statistically significant difference in observer performance between AMBER (Area under the ROC curve AZ = 0.934) and conventional radiography (AZ = 0.868) in the task of detecting diffuse lung disease.

The Random Walk

Consider the stochastic process which is the path of a particle which moves along an axis with steps of one unit at time intervals also of one unit. Suppose that the probability is p of any step being taken to the right, and is q = 1 - p of being to the left. Suppose also that each step is taken independently of every other step. Then this process is called the unrestricted random walk. If the particle is in position 0 at time 0, determine the probability that it will be in position k after n steps.

The moments of forward recurrence time

Abstract The moments of the forward recurrence time of an ordinary renewal process are derived in terms of the renewal function and the moments of the common lifetime distribution. The covariance between the forward recurrence time and the number of renewals is also obtained. Asymptotic formulae as the process is allowed to run on for a fixed long time are given.

Size determination of transparent spheres in an opaque specimen from a slice

A practical procedure is described for estimating the size distribution of transparent spherical particles embedded in an opaque specimen from the sizes of the profiles in a thin slice of the specimen. This complements Rose (1980), the case of opaque spheres in a transparent medium.

The Stereological Analysis of Two-Phase Particles

Three stages in the stereological analysis of two-phase particles are identified. These are illustrated by (a) random probes and sections through a sphere with a spherical nucleus, (b) random probes through discs with a chord as phase interface, and (c) random probes and sections through a population of right cylinders with convex cross-sections and perpendicular cross-section phase interfaces.

The construction of invariant random paths through three-dimensional specimens

A quick way of constructing invariant random (IR) paths (sometimes called IUR or μ‐random) through three‐dimensional specimens is described. It uses the result that the secant joining two points chosen uniformly and independently on the surface of a sphere is IR.

The superposition of the backward and forward processes of a renewal process

A fixed sampling point O is chosen independently of a renewal process on the whole real line. The distances Y1, Y2, ... from O to the renewal points of , when they are measured either forwards or backwards in time, define a point process . The process is a folding over of the past of onto its future. It is the superposition of two equilibrium renewal processes which are known to be independent only when is a Poisson process. The joint distribution of Y1, Y2, ..., Yk is found. The marginal distribution of 2Yk is shown to be the same as that of the distance from O to the kth following point of . The intervals of are shown to have a stationarity property, and it is proved that if any pair of adjacent intervals of are independent, then is a Poisson process.

Continuous Univariate Distributions.

8. Continuous Univariate Distributions, vol. 1, 2nd edn. By N. L. Johnson, S. Kotz and N. Balakrishnan. ISBN 0 471 58485 9. Wiley‐Interscience, New York, 1994. xx + 756 pp. £66.

Probability Theory, An Analytic View.

21. Probability Theory, an Analytic View. By D. W. Stroock. ISBN 0 521 43123 9. Cambridge University Press, Cambridge, 1994. xvi + 512 pp. £30 (