Stochastic models for a number of reversible chemical reactions involving one or two second-order steps are examined and exact solutions which involve standard orthogonal polynomials are found for the equilibrium states. The expected values and variances of the numbers of molecules present in the systems at equilibrium are expressed in terms of these polynomials. Comparison of the stochastic mean value and the classical deterministic expression, shows that they are equivalent for most systems of physical interest, that is when large numbers of molecules are present. The relative fluctuation is found to be O (N−½), where N is of the order of the number of molecules present in the system, in agreement with the results of equilibrium statistical mechanics.

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Consideration of the number of events in a Poisson process in excess of a threshold vslue r, when it is assumed that at least T events do occur, leads to the Displaced Poisson Distribution. The recurrence relationship between successive probabilities is a simple generalization of that of the Poisson distribution. Accordingly the fitting of the distribution to data is straightforward, using either a desk machine OF computer.It is a highly flexible unimodal distribution catering for both under-and over-dispersed data and generally fits Poisson type data well. Its properties have been discussed, moment and maximum likelihood estimators for the cases, r known and unknown have been given, together with several numerical examples. D

Suppose that a new hospital is to be planned or an existing one reorganized according to the modern Progressive Patient Care System (Hartley et al. (1968)). In this system, there are three types of wards, viz.