On the optimal mapping of distributions

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We consider the problem of mappingXY, whereX andY have given distributions, so as to minimize the expected value of ∣XY2. This is equivalent to finding the joint distribution of the random variable (X, Y), with specified marginal distributions forX andY, such that the expected value of ∣XY2 is minimized. We give a sufficient condition for the minimizing joint distribution and supply numerical results for two special cases.

Communicated by D. Q. Mayne