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I would love to answer another question on this site, but I am totally unfamiliar with the required technique. I mean, I don't even know the sub-field's name.

The field I am looking for is one that is concerned with the optimisation of (probability or cumulative density) functions w.r.t. some integral.

What is the appropriate (nearest) field?

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The calculus of variations is extremely relevant to your problem. If you ahve constrints, then you will need to also know the method of Lagrange Multipliers. This work ties it all together

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    $\begingroup$ Thanks. 1) I was hoping that the restriction that the sought functions must be probability density functions might yield a more simple analysis. And 2) I think (but I'm quite unsure) that my problem doesn't require smoothness, whereas CoV does? I.e., the space of all pdfs/cdfs on $[0,1]$ doesn't seem appropriate for CoV. Am I wrong? Any further guidance? $\endgroup$ – Keep these mind Nov 8 '13 at 17:26
  • $\begingroup$ CoV methods (with Lagrange multipliers) has been used extensively in proability and statistis. Look up Maximum Entropy Distributions for examples. I don't have an answer regarding smoothness. I guess it doesn't include (or won't examine the space that includes) the more esoteric continuous distributions like the Cantor Distribution. Can you live with that? Also, the space of ALL CDFs includes on [0,1] includes all monotonic functions that go from 0 to 1. I don't know of a theory that will explore such a general space. $\endgroup$ – user76844 Nov 8 '13 at 17:57
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You might be looking for optimal transport theory http://en.wikipedia.org/wiki/Transportation_theory_%28mathematics%29

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