When considering the joint distribution of two specific, identically distributed random variables (labelled X and Y) for different assumed correlations, I noted that a two parameter family of feasible joint distributions formed a convex set. How general is this observation?
The specific discrete variable I was considering takes values 0, 0.5 and 1.0 with probabilities 0.22, 0.30 and 0.48 respectively.Using a natural notation of p(i,j) as the probability that X takes value i and Y takes value j (i=1 means X=0, j=1 means Y=0 etc )led to joint distributions of the form: -0.56 + p(2,2)+p(3,3) 0.3-p(2,2) 0.48-p(3,3) 0.30 - p(2,2) p(2,2) 0 0.48 - p(3,3) 0 p(3,3). Non-negativity conditions produce linear constraints in terms of p(2,2) and p(3,3). Moreover since X and Y have an assumed fixed distribution their correlation is also a linear function of p(2,2) and p(3,3). For a correlation of 0.1 two alternative p(2,2), p(3,3) values are ( 0.3, 0.3377) and (0.269,0.3455). Comments and references to relevant lieterature would be a great help. Thanks Ian Calvert Ian Calvert . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
