In genetics one frequently encounters variates which behave, to a close approximation, as diffusion variates with drift and diffusion coefficients m(x),v(x), respectively, where after a suitable rescaling of the time axis, m(x) and v(x) are of the form

In this paper we shall be concerned with two mathematical models of infinite dams. In the first model independent random inputs occur at regular time intervals and in the second model independent random inputs occur in accordance with a Poisson process. The first model has already been studied by Gani, Yeo and others, and the second model by Gani and Prabhu, Gani and Pyke, Kendall, and others. For both models we shall find explicit formulas for the distribution of the content of the dam and that of the lengths of the wet periods and dry periods. The proofs are elementary and based on two generalizations of the classical ballot theorem.

In a ballot, candidate A scores a votes and candidate B scores b votes. Suppose the ballots are drawn out one at a time, and denote αr and βr the number of votes registered for A and B, respectively, among the first r votes recorded. Further, let Δa,b be the number of subscripts r satisfying the strict lead condition , let be the number of subscripts r satisfying the weak lead condition ; and suppose all possible () voting records are equally probable. The probability distributions of the number of strict and weak lead positions corresponding to and , respectively, have been determined in [4] for a≧b.

The probability theory of storage systems formulated by P. A. P. Moran in 1954 has now developed into an active branch of applied probability. An excellent account of the theory, describing results obtained up to 1958 is contained in Moran's (1959) monograph, Considerable progress has since been made in several directions-the study ofthe time-dependent behaviour ofstochastic processes underlying Moran's original model, modifications of this model, as well as the formulation and solution of new models. The aim of this paper is to give an expository account of these developments; a comprehensive treatment will be found in the author's forthcoming book [Prabhu (1964)].

Smoluchowski's classical analysis of the temporal fluctuation, under diffusion equilibrium, of the number of particles in a fixed region R of space is generalised to a set of disjoint regions; specifically, the single Smoluchowski region is divided into a finite number of non-intersecting subregions. The generalisation allows a more rigorous test of some of the consequences of the Einstein-Smoluchowski theory of Brownian motion to be carried out, and at the same time enables the Avogadro constant to be estimated with greater precision than is possible with the single region. In particular, the reversibility paradox of Loschmidt and the recurrence paradox of Zermelo are reexamined from the point of view of the fluctuation of configurations (a configuration being defined as the set of occupation numbers for the various subregions) rather than that of total concentration for the single region.

The purpose of this paper is to construct a theory of the amount of information provided by an experiment which does not rely on what Good (1962) has termed the modern Bayesian principle that it is legitimate to use the axioms of probability even when this involves the use of probabilities of hypotheses. In this respect the theory of this paper differs from the Lindley (1956), Mallows (1959) and Good (1960) each of which is written from a Bayesian viewpoint. Lindley (1956) expresses the opinion that Bayesian ideas would seem to be necessary to the development of a theory of the amount of information provided by an experiment and it is of interest therefore to determine how far such a theory may be developed without Bayesian ideas. There is further a need to use such a theory to examine how prior knowledge can be expressed quantitatively, and used in accordance with that theory.