Binomial and Negative Binomial Analogues under Correlated Bernoulli Trials

Article excerpt

1. INTRODUCTION

A daily customer has to choose between two brands, A and B, of a given product. A model for representing the customer switching patterns discussed by Olkin, Glesser, and Derman (1980, pp. 459-460) assumes that only the customer's choice of brand on the immediately preceding day, day i, affects his or her choice on day i + 1. Suppose that when the customer chooses Brand A on one day, then he or she will choose Brand A the next day with probability .8; when the customer chooses Brand B on one day, then he or she will choose Brand A the next day with probability .3. The customer chooses a brand randomly the first day he or she buys the product. The following are questions of concern to the product makers.

[Q.sub.1]: How likely is that the customer will choose Brand A on 20 days out of the first month (30 days) he or she buys the product?

[Q.sub.2]: How likely is it that the twelfth time the customer chooses a Brand A product occurs in 15 days?

If the probability of choosing each brand was constant from day to day, then any student with an elementary course on probability could provide an answer to the aforementioned questions by invoking the binomial and negative binomial distributions because the chain of brand choices across the various days will simply be a sequence of independent Bernoulli trials. In the present application. however, the trials are not independent, and the answers to [Q.sub.1] and [Q.sub.2] demand careful examination.

The problem just described is an example of Markov-correlated Bernoulli trials. These correlated trials find application in many applied disciplines. For instance, Estes (1950) and others since then have used binary Markov models to represent learning (see Olkin et al. 1980, pp. 449-450). An application to binary signal transmission in communications was discussed by Pfeiffer and Serum (1973, p. 358). Kemeny and Snell (1960, p. 31) discussed a model for representing gambler's behavior in choosing between two slot machines. An application to model the starting-up reliability of power-generation equipment operated on gas (e.g., lawnmowers) is presented in Viveros and Balakrishnan (1993). Dry--rainy weather patterns (Ross 1993, p. 138) have also been modeled with Markov-correlated Bernoulli trials. An interesting application to modeling vowel--consonant patterns in biblical text, and more generally in languages, was made by Newman (1951). Among the eight languages investigated, Newman (1951) found that Samoan and Lifu (the largest of the Loyalty Islands, a group located in the south Pacific Ocean east of New Caledonia) are the only languages whose vowel--consonant patterns appear to follow first-order Markov chains. King James English has patterns that appear to follow a third-order chain.

To examine this class of problems more generally, let [p.sub.0], [p.sub.1], and [p.sub.2] be [p.sub.0] = Pr([O.sub.1] = S), [p.sub.1] = Pr([O.sub.i] = S | [O.sub.i-1] = S), and [p.sub.2] = Pr([O.sub.i] = S | [O.sub.i-1] = F), where [O.sub.i] is the outcome of trial i (i [is greater than or equal to] 1), and S and F denote success and failure, respectively. Let [q.sub.j] = 1 - [p.sub.j] (j = 0, 1, 2).

This simple correlation structure is tantamount to a stationary two-state Markov chain with state space {S, F} and transition matrix with rows ([p.sub.1], [q.sub.1]) and ([p.sub.2], [q.sub.2]). Therefore, any situation giving rise to such Markov chains will provide an application for the results discussed in this article (e.g., see Kemeny and Snell 1960).

This article centers merely on probabilistic considerations: we do not discuss statistical issues such as the estimation of [p.sub.0], [p.sub.1], and [p.sub.2] and the testing of the Markovian assumption. We refer the interested reader to the work of Bishop, Fienberg, and Holland (1975, chap. 7) and Cox and Snell (1989, pp. 98-105) for the statistical analysis of binary Markov chains and time series. …