Volume 28 - Issue 2 - December 2003

The oscillation of a loaded spring

R. Champion and W. L. Champion


We examine some aspects of the oscillatory motion of a mass M suspended on an ideal spring. Departure from simple harmonic oscillation, due to the contribution of normal modes of oscillation with frequencies greater than the first normal mode, is modelled using a simple wave equation, which takes account of the mass of the spring as well as M. We compare theory with data obtained using standard laboratory equipment, and provide an explanation of the oscillatory discrepancies from simple harmonic motion.

Some unusual birth-and-death processes

P. R. Parthasarathy


Transition probabilities of certain birth-and-death processes are presented using their associated continued fraction representations.

On the means of deterministic and stochastic populations

Jennifer Switkes, Stephen Wirkus, Ioana Mihaila and Randall Swift


Familiar results for deterministic birth−death−immigration−emigration (BDIE) population models are compared with the expected population sizes predicted by a related stochastic BDIE model. Although under standard assumptions these two results are not equivalent, we show that by removing restrictions that explicitly prohibit negative population sizes, the two models are reconciled.

Transient solution of an M/M/2 queue with catastrophes

B. Krishna Kumar and S. Pavai Madheswari


In this paper, we derive the transient probabilities for an M/M/2 queue with the possibility of catastrophes at the service stations. The steady-state probabilities of the system size are also obtained. The asymptotic behaviour of the probability of the server being idle and the mean system size are discussed. The steady-state mean of the system size is also studied and numerical illustrations are provided.

Optimal sequential selection from among n boxes

Nelson M. Blachman


There are n boxes with identically distributed, statistically independent contents. You look in one of them and may keep its contents if sufficiently attractive. Otherwise you may forever forego it and may look in another, etc., stopping with the first contents that meet a suitable criterion. The strategy that maximizes the expected utility of the outcome of this selection process for a given probability distribution of the contents is applied to the case where the utility function is the quantile rank. For this utility function, attention is then given to the nonparametric case where the probability distribution is entirely unknown in advance. Here it turns out to be best to examine the first quarter of the boxes before beginning to accept any contents that exceed an optimal threshold, the expected quantile rank of the outcome being equivalent to that from approximately n/2 boxes with known probability distribution.

The Collins case again

Halvor Mehlum


This note follows up Turner's (2002) discussion of the Collins case. We derive some results which correct an oversight in Turner's paper. The discussion illustrates the general insight that it is important for the analysis of statistical data to know exactly how the data were collected.

On the central moments of the multidimensional Gaussian distribution

Kostas Triantafyllopoulos


This paper gives a simple derivation of the order-k central moments of the multidimensional Gaussian distribution. After a brief review of the literature, the proof follows using basic induction. Thus, a problem which required advanced mathematics can now be treated in an elementary way, and can hence be more accessible to scientists and practising statisticians.

Sample variance and first-order differences of observations

Anwar H. Joarder


It is proved that the variance of a sample can be calculated from first-order differences of its observations. It can also be represented by a quadratic form in the differences of the observations via a constant matrix (depending on the sample size) which is open for further study. Avoiding use of the mean in the calculation of the variance is expected to increase precision, especially if computer programs are used. An alternative method is also presented for the calculation of the variance from a frequency distribution.