We develop some approaches to the characterization of distributions of real-valued random variables, useful in practical applications, in terms of conditional expectations and hazard measures. We prove several representation theorems generalizing earlier results, and establish stability theorems for two general characteristics introduced in this paper.

In the incompatibility system operating in Lythrum salicaria, the three incompatibility types Long, Mid and Short will be equally frequent in a population of infinite size. But in small populations the complete system cannot be maintained indefinitely because of random loss of incompatibility types. A simulation model for such populations shows that this may be an important factor for populations consisting of less than 20 plants. When loss actually occurs, it will in most cases involve the Short type and in some cases the Mid type, but only rarely the Long type. This result may explain why the Long type appears to be the most common one in nature and Short the most infrequent one.

One general and three specialized models of the Bush–Mosteller type are presented to describe the kind of learning experiment where the response of the learner is always reinforced. Inhomogeneity is admitted. The random sequences of response probabilities and of responses associated with the different models are considered. Information about the existence and the distribution of asymptotic response probabilities is provided. The stress is on sufficient and necessary conditions for convergence (a.s. or with positive probability) of the response sequence, which is what ‘learning' means.

The treatment of selfing in genetic algebras [3] can be extended by the definition of an analogous operator on the copular algebra which corresponds to brother–sister mating, one on the duplicate of the copular algebra corresponding to mating between double first cousins, etc.

Suppose, to take a simple example, we have a set of sibships. Each individual (sib) has a certain measured character x. We wish to estimate the interclass (or sib–sib) correlation for this character. Our model is as follows. In sibship s the observed values x are supposed drawn from a distribution with (true) mean ms and variance vs. The sibships are drawn from a population of sibships, so that the ms have a mean M and variance B, and the vs have mean A. (We also suppose that the number ns of sibs in sibship s is independent of ms and vs.) The interclass correlation is then easily seen to be

ρ = B/(A + B).

The obvious way to proceed is to estimate A and B first, and then find ρ. Take a sample of k sibships. In sibship s the observed mean is and the deviance (sum of squares of deviations) δs. Let n = σ ns = the total number of individuals in the sample, and δ = σ δs = the total deviance within sibships. Proceed iteratively as follows:

Consider a decision-maker who is in charge of a number of activities. At each of a sequence of decision points in time, he selects—on the basis of their performance—the set of activities to be continued further and allocates his limited resources among them. Activities receiving larger allocations tend to improve their performance, while others receiving smaller allocations tend to deteriorate. We present a controlled random walk model for the progress of these activities. The problem of maximizing the net infinite horizon discounted return is formulated in the framework of Markov decision theory, and existence of optimal strategies established. It is shown that both the optimal selection and allocation strategies exhibit a ‘favoring the leaders' behavior. Finally, explicit solutions to certain special cases are obtained illustrating these results.

One of two random variables, X and Y, can be selected at each of a possibly infinite number of stages. Depending on the outcome, one's fortune is either increased or decreased by 1. The probability of increase may not be known for either X or Y. The objective is to increase one's fortune to G before it decreases to g, for some integral g and G; either may be infinite.

In Part I (Berry and Fristedt (1980)), the distribution of X is unknown and that of Y is known. In the current part, it is known that either X or Y has probability α of increasing the current fortune by 1 and the other has probability β of increasing the fortune by 1, where α and β are known, but which goes with X is not known. We show that optimal strategies exist in general and find all optimal schemes when α = 0 and when α + β = 1. In both cases myopic strategies are shown to be optimal. A counterexample is used to show that myopic strategies, while intuitively very appealing, are not optimal for general (α, β).

Consider a single gene locus for which the associated character is not sex-linked, and suppose that the character is exhibited as one of alleles. Under the Mendelian model of inheritance, if genotype frequencies are constant, the distribution of zygotes is stable from the second generation on, regardless of the value of n. (This is the celebrated Hardy–Weinberg law—see Hardy (1908).) In 1924 Bernstein raised the question of whether the Mendelian model is, in fact, a necessary consequence of the assumption that the zygote distribution is stable from the second generation on; when n = 2 or 3 Bernstein gave an affirmative solution, using elementary but tedious manipulation of probabilities. Lyubich (1971) used methods of convex analysis to obtain a partial solution of Bernstein's problem for general n. Holgate (1975) introduced algebraic methods in order to present a clearer picture of the underlying structures, and to relate Bernstein's problem to the algebras studied by Etherington (1939), Schafer (1949) and others. The purpose of this paper is to expound recent work of the author and others, building on Holgate's and Lyubich's ideas. The subject-matter of the paper is contained in a forthcoming joint paper by the author and A. E. Stratton.

The probability distribution of the heterogenic (non-identical by descent) fraction of the genome in a finite monoecious random mating population has been derived. It was assumed that in any generation both the lengths of heterogenic and homogenic segments are exponentially distributed. An explicit expression is given for the expected number of ‘external junctions’ (sites that mark the end of a heterogenic segment) per unit map length in any generation. The latter necessitates the introduction of two higher-order identity relations between three genes, and their recurrence relations. Theoretical results were compared with the outcome of a series of simulation runs (showing a very good fit), as well as with the results predicted by Fisher's ‘theory of junctions’. In contrast to Fisher's approach, which only applies when the average heterogeneity is relatively small, the present model applies to any generation.

The problem of controlling input to a stochastic input-output system by accepting or rejecting arriving customers is analyzed as a semi-Markov decision process. Included as special cases are a GI/G/1 model and models with compound input and/or output processes, as well as several previously studied queueing-control models. We establish monotonicity of socially and individually optimal acceptance policies and the more restrictive nature of the former, with random rewards for acceptance and both customer-oriented and system-oriented non-linear waiting costs. Distinctive features of our analysis are (i) that it allows dependent interarrival times and (ii) that the monotonicity proofs do not rely on the standard concavity-preservation arguments.