Abstract
In a first course on probability and distribution theory, when bivariate distributions are discussed, the trinomial distribution is presented as an example of a bivariate distribution with binomial marginals. Later, the student encounters Aitken and Gonin's (1935) bivariate binomial distribution arising from fourfold sampling with replacement (c.f. Kendall and Stuart, 1963, p. 141). It is unfortunate, however, that textbooks do not relate the trinomial distribution to Aitken and Gonin's bivariate binomial distribution. This may lead to the misunderstanding that the two distributions are bivariate binomial distributions which are different in structure. This misunderstanding grows further when students are introduced to the canonical expansion of Aitken and Gonin's bivariate binomial probability function, that is, as a series bilinear in Krawtchouk's polynomials. A student would think that the trinomial distribution, which is very simple in nature, cannot be related to something as complex as Krawtchouk's polynomials. The aim of the present note is to point out the fact that the two distributions have similar structures, and in fact the trinomial distribution is a special case of Aitken and Gonin's bivariate binomial distribution. Hence, the canonical structure of the trinomial distribution is indicated. Aitken and Gonin's (1935) bivariate binomial is based on a population in which each individual may be classified as either A or A and at the same time as either B or B. Each individual in a sample of size n is then classified in one of the four mutually exclusive classes AB, AB, AB, AB with probabilities pit, Pto, Pox, Poo, respectively. If X1 is the number of occurrences of A and X2 is the number of occurrences of B, then the marginal distributions of Xx and X2 are binomial (n, pt) and binomial (n, P2), where
Published Version
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