Timeline for Bound on the joint distribution of three real random variables with given two dimensional marginals
Current License: CC BY-SA 3.0
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| Aug 19, 2016 at 3:12 | comment | added | user44143 | I think the discrete approach can lead to a solution, eg: Start with X,Y,Z independently distributed as B(6,1/2), linearly transformed so each has mean 0 and variance 1. Restrict these to octants with XYZ>0. Then experiment over all prisms [a,b]x[c,d]x[e,f] to find where moving mass from one set of four vertices to the other will most increase E(XYZ). Repeat. The B(3,1/2) case leads to the solution above; from the B(6,1/2) case we could probably guess the answer with continuous setup easily enough. | |
| Aug 18, 2016 at 2:16 | comment | added | Jean Duchon | The discrete case is interesting too, though it doesn't address the question proper. Why not consider simply $X=\pm1$ with probability 1/2 each, where the solution is trivially $\frac14(\delta_{+++}+\delta_{+--}+\delta_{-+-}+\delta_{--+})$ ? That makes $XYZ=1$ with probability 1 ... | |
| Aug 17, 2016 at 3:42 | history | edited | user44143 | CC BY-SA 3.0 |
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| Aug 17, 2016 at 3:17 | history | edited | user44143 | CC BY-SA 3.0 |
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| Aug 17, 2016 at 3:03 | history | answered | user44143 | CC BY-SA 3.0 |