On a Simple Construction of a Bivariate Probability Function With a Common Marginal

Abstract

We introduce a family of bivariate discrete distributions whose members are generated by a decreasing mass function p, and with margins given by p. Several properties and examples are obtained, including a family of seemingly novel bivariate Poisson distributions.

• Bivariate distributions
• Covariance
• Discrete distributions
• Mixtures of Poisson
• Poisson
• Probability generating function.

Notes

The common pmf is given by for , with being the Gauss hypergeometric function. For a ⩾ 1, we can show that p decreases on and that such a p generates the joint probability mass function in (2(2) ) via Lemma 1.

A multivariate generalization, which is not pursued here, for a joint distribution for (X₁, …X_n) generated from a nonincreasing p, with univariate marginals given by p, as above is of the form

or by a direct evaluation

This is not a necessary condition. For instance, the negative binomial distributions presented earlier in the table are Poisson mixtures with Gamma distributed α.

By taking a sequence of distributions with mean r converging in distribution to a Bernoulli(r).

Note that S is not necessarily unique but it does not matter in such cases which choice is made.