Distributions with normal conditionals

Abstract

If (X, Y) has a classical bivariate normal distribution with density$fx,y(x,y) = {1 \over {2\pi \sigma 1\sigma 2\sqrt {1 - \mathop p\nolimits^2 } }}\exp \{ - {1 \over {2(1 - \mathop p\nolimits^2 )}}[({{x - \mu _1 } \over {\sigma _1 }})2 - 2p({{x - \mu _2 } \over {\sigma _1 }}) + ({{y - \mu _2 } \over {\sigma _2 }})2]\} ,$(3.1)then it is well known that both marginal densities are univariate normal and all conditional densities are univariate normal. In addition, the regression functions are linear and the conditioned variances do not depend on the value of the conditioned variable. Moreover the contours of the joint density are ellipses. Individually none of the above properties is restrictive enough to characterize the bivariate normal. Collectively they do characterize the bivariate normal distribution but, in fact, far less than the complete list is sufficient to provide such a characterization. Marginal normality is of course not enough.