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I have two random variable X and Y. I know that $$E_X\left[X \log \frac{X}{e}\right] < E_Y\left[Y \log \frac{Y}{e}\right]$$

Using the above relation can I say anything about the relation between $E_X[X]$ and $E_Y[Y]$ ?

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  • $\begingroup$ Is that all you know about these random variables? $\endgroup$ – JohnK Feb 4 '16 at 20:32
  • $\begingroup$ If needed, we can assume that the random variables X and Y are independent. $\endgroup$ – neo89 Feb 4 '16 at 20:59
  • $\begingroup$ Since you only ask about marginal expectations, independence is irrelevant. $\endgroup$ – kjetil b halvorsen Sep 26 '17 at 18:01
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I will only give some hints.

Clearly, $X>0$ is a positive random variable for this to make sense. Note that $x \log(x/e) = x\log x - x\log e = x\log x -x$. Define $f(x) = x\log x -x$. So your claim becomes $\DeclareMathOperator{\E}{\mathbb{E}} \E_X f(X) < \E_Y f(Y)$. Calculate the two first derivatives of $f$, and conclude that $f$ is convex. Then: Jensen inequality: https://en.wikipedia.org/wiki/Jensen%27s_inequality

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