The kin number problem in its simplest form is that of the relationship between sibship sizes and offspring numbers. The fact that the distributions are different, and the relationship between the two, is well known to demographers. It is important in such applications as estimating fertility from sibship rather than offspring counts. Further studies have been made, concerning relatives of other degrees of affinity than siblings, but these did not usually yield joint distributions. Recently this aspect of the problem has been studied in the framework of a Galton–Watson process (Waugh (1981), Joffe and Waugh (1982)).

In these studies the population is treated as monotype. Applications such as pedigree studies of diseases require a multitype approach (in the example, two types: victims and others). In this paper such a study is undertaken. The existence of more than one type complicates the time-reversal used in the previous studies, and raises questions about the way in which the focal individual (called ‘Ego') is sampled. These are dealt with, and joint distributions obtained, under a number of sampling schemes which might arise naturally.

The kin number problem involves the relationship between sibship sizes and offspring numbers, and also numbers of relatives of other degrees of affinity of a random member of a population, to be called Ego. The problem has been well known to demographers for some time, but results obtained only gave expected numbers. Recently a study of it, based on the Galton-Watson process, was made, with a view to obtaining joint distributions (Waugh (1981)). In the latter study it was assumed that the population was large, and thus some of the results obtained were approximations.

In the present paper exact distributions are obtained, for any size of population. This can be of use in applications, where the population considered may be a small, isolated tribe or other special group. As a theoretical investigation, it replaces some heuristic arguments with limiting properties that are intrinsic to the process and it makes it possible to evaluate the previous approximations.

The kin number problem concerns the relationship between the distribution of the number of offspring of a randomly chosen individual in a population, and that of the number of relatives of various degrees of affinity of a randomly chosen individual (referred to as ‘Ego’). This problem is considered in terms of a population consisting of a large number of simultaneously developing Galton-Watson family trees. By time-reversal starting with the epoch at which Ego is sampled, it is shown that the number of offspring of Ego's parent (Ego plus siblings) and the number of offspring of Ego's grandparent (Ego's parent plus Ego's parent's siblings) have the same distribution, and the probability generating function (p.g.f.) is obtained in terms of the reproduction p.g.f. Further development of the method yields the joint p.g.f. of any number of generations prior, and subsequent, to that of Ego. Means and variances are obtained, and numerical examples are given based on data for the reproduction p.g.f. in two contrasting human populations. Various applications including demography and cancer research are discussed.