Correlation between the numbers of two types of children in a family, using the Markov-Polya model

Abstract

Biologists, human geneticists, demographers, statisticians, and social scientists have long been concerned with questions relating to the family size and the distribution of boys and girls within families of different sizes. One of the significant questions raised by such studies, whether or not there is a correlation between the numbers of children of each sex, has been examined by Rao et al. (1973). Assuming a binomial model, these authors developed a method of computing the correlation between two types of children in a family. If the sibship size N follows a negative binomial (or binomial), Rao et al. (1973) have shown that the correlation ϱ between the numbers of boys, X, and girls, Y, in a family is positive (or negative), and when N has the Poisson distribution, ϱ is zero. In the present paper, the same question is examined under the Markov-Polya model. General expressions for the correlation coefficient, ϱ, are obtained when N has (1) a generalized power-series distribution, and (2) a modified power-series distribution. Certain characterization theorems are proved in the context of correlation between X and Y. Thus, it is shown that (a) under the Markov-Polya distribution with positive (negative) contagion, ϱ is zero if, and only if N has a negative binomial (binomial) distribution; (b) when N has the Poisson distribution, ϱ is greater or less than zero according as the parameter of contagion is negative or positive.