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Optimization : Comparing two probability distributions.

F and G are cumulative probability distributions with identical support. G first order stochastically dominates F, i.e., for every X on the support, F(x) > G(x). Prove (or disprove) the proposition that argmax [X(1-G(X))] > argmax [X(1-F(X))], where argmax is the value of x that maximizes the expression in brackets.

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Solution Preview

Here is an example demonstrating that the proposition is WRONG.
(Proof of wrong can be done by a single example to the contrary of a proposition)

Let G(x) = x for all x in [0,1] ;
Then arg(max(x(1-G(x)))) = 1/2

Let ...

Solution Summary

Two probability distributions are compared.