Distributions with Normal Conditionals

Abstract

At first glance, it is surprising that there can exist bivariate distributions with all their conditional densities of the normal form, which are not the classical bell-shaped bivariate normal that we are so familiar with. But such distributions do exist and have actually been at least partially known and understood for more than 50 years. After reviewing the history of such distributions we focus on developing a convenient parametric representation of all such normal conditionals distributions. The role of the classical bivariate normal distribution as a special case is investigated in some detail. In fact we begin our discussion with this topic.