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## Main Question or Discussion Point

I've encountered this question in some theoretical work I've been doing. Suppose you have two pdfs fH(x) and fL(x) with the same support (bounded or infinite... you can assume whatever you want about it).

Suppose that the expected value of a random variable with pdf fL(x) is greater than the expected value of a random variable with pdf fH(x).

Is it true that the integral over the common support INT{ (fH(x)^2)/fL(x) } dx is larger than one? Is it true if the CDF FH(x)<=FL(X)?

I've checked family of distributions x^a, exponential, Normal, binomial and the claim is true in these classes, but I dont know if its true in general....

Suppose that the expected value of a random variable with pdf fL(x) is greater than the expected value of a random variable with pdf fH(x).

Is it true that the integral over the common support INT{ (fH(x)^2)/fL(x) } dx is larger than one? Is it true if the CDF FH(x)<=FL(X)?

I've checked family of distributions x^a, exponential, Normal, binomial and the claim is true in these classes, but I dont know if its true in general....