A family of bivariate distributions generated by the bivariate Bernoulli distribution. (English) Zbl 0575.60023

Marshall, Albert W.; Olkin, Ingram

J. Am. Stat. Assoc. 80, 332-338 (1985).

If \(X_ 1,X_ 2,..\). is a sequence of independent Bernoulli random variables, the number of successes in the first n trials has a binomial distribution and the number of failures before the rth success has a negative binomial distribution. From both the binomial and the negative binomial distributions, the Poisson distribution is obtainable as a limit. Moreover, gamma distributions (integer shape parameters) are limits of negative binomial distributions, and the normal distribution is a limit of negative binomial Poisson, and gamma distributions. These basic facts from elementary probability have natural extensions to two dimensions because there is a unique natural bivariate Bernoulli distribution. In this article, such extensions yielding a family of bivariate distributions are obtained and studied.

MSC:

60E99	Distribution theory
60F05	Central limit and other weak theorems

Keywords:

bivariate hypergeometric distribution; bivariate exponential distribution; binomial distribution; negative binomial distribution; gamma distributions