Examples of Decompositions of Chi-Squared Variables

Two examples of absolutely continuous bivariate distributions are given.The first example illustrates the fact that the sum of two random variables can be .2, one of the variables .2, the other variable positive but not necessarily .2.The second example illustrates the fact that the sum of the variables can be .2, each variable can be .2, the degrees of freedom add up properly but the two variables need not be independent.