Construction of Certain Bivariate Distributions

Abstract

The purpose of this note is to construct bivariate distributions which will illustrate the following situations which are frequently mentioned in textbooks but handy examples of which are not obvious to come by. (i) If X and Y are two random variables with the moment generating functions Ml (t) and M2 (t), respectively, then independence between X and Y implies that the moment generating function M(t) of X + Y is Ml(t)M12(t). However, M(t) = Ml(t)M2(t) does not imply independence between X and Y. An example of this is given in Cramer [1946], p. 317. (ii) Univariate marginal distributions of a bivariate normal distribution are normal but you could have a bivariate non-normal distribution whose univariate marginals are normal. In general, it may be possible to use the present method to construct a fairly arbitrary bivariate distribution with given marginals. (iii) The joint distribution of two non-independent random variables X and Y is such that X2 and Y2 are independent. (Parzen [1960], p. 297). To begin with let fi(x) and f2(y) be the probability densities of the two continuous random variables which are symmetric about zero and with the moment generating functions M1 (t) and M2 (t), respectively. Let g(., .) be a real valued, integrable function of two variables such that I g(x, y) I < 1 for all real x and y. Consider a two dimensional random variable (X, Y) with the joint density given by