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Volume 29 - Issue 2 - December 2004
S. P. Decent, J. C. Wathan and S. E. Brown
This paper describes a feasibility study on the use of mathematical models in social research, and is particularly aimed at a type of problem which is normally tackled using statistical methods. We describe higher education entry in the UK and compare the results of the model to University and Colleges Admissions Service (UCAS) data. In particular we aim to understand the age distribution of new students in higher education. The results are very encouraging and suggest that mathematical models potentially yield much wider benefits for the social sciences than is currently appreciated by the social science community. We discuss briefly the possible role of mathematical modelling in the social sciences, and suggest how it might complement the statistical methods more commonly used by social scientists.
John C. B. Cooper
This paper provides an introduction to the theory of cointegration analysis and demonstrates an application by testing the famous purchasing power parity theory from international economics.
Robert C. Dalang and Violetta Bernyk
We propose a mathematical model for the TV game show ‘Who Wants To Be A Millionaire?’ Using stochastic optimization methods, we obtain the optimal strategy which maximizes the player's expected payoff. The model provides answers to questions such as ‘what can the player expect to win?’ and ‘what are the chances of winning a million dollars?’. This optimal strategy is presented in a simple form.
Yoshifusa Ito and Hiroyuki Izumi
We show that the exact distribution of the sample correlation coefficient can be obtained by a geometric method from the joint probability density function of the sample standard deviations and the sample correlation coefficient. This joint probability density function was obtained by Fisher using a geometric method. Thus, if our method is combined with Fisher's, then the exact distribution of the sample correlation coefficient can be obtained by using geometric methods only.
A. Yu. Mitrophanov
For a finite continuous-time Markov chain, we prove a sufficient condition for reversibility in terms of a spanning tree and the corresponding fundamental cycles of the chain's transition graph. We demonstrate how this sufficient condition can be used to construct reversible Markov chains given the transition graph.
We model an unbiased random walk with catastrophe. Unit increases and decreases occur at equal rates. Catastrophes, which return the process to state zero, occur at a distinct constant rate. The variance is computed for this process, which due to symmetry has expected value zero. The stationary distribution is determined. Finally, transient probabilities are computed recursively using Laplace transforms and these probabilities are investigated numerically.
Saralees Nadarajah and Samuel Kotz
A truncated version of the Student t distribution is introduced. Unlike the t distribution, this possesses finite moments of all orders and could therefore be a better model for certain practical situations. Two such situations are discussed. Explicit expressions for the moments of the truncated distribution are also derived.