Saturday, 19 November 2016

An inequality involving two probability densities

I cannot prove the following inequality, which I state below:



Let p,q be two positive real numbers such that p+q=1. Let f and g be two probability density functions. Then, show that:



Rp2f2+q2g2pf+qgp2+q2 .



I tried to use Cauchy-Schwarz and even Titu's lemma, but got nowhere. Any help will be greatly appreciated. Thanks!

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