# Shannon entropy and inequality of expectations

Consider two distinct probability distributions $P(X)$ and $Q(Y)$---defined on the same domain---with (Shannon) entropy of $H(X)$ and $H(Y)$. I am interested to prove that $$H(X) \leq H(Y) \implies \sum_{x}\sum_{x'} P(x)P(x')|x-x'| \leq \sum_{y}\sum_{y'} Q(y)Q(y')|y-y'|$$

• Why do you believe this would be true? What kind of examples have you looked at? Jun 1 '15 at 17:17

$$P(0) = \tfrac{1}{2}, \quad P(1) = 0, \quad P(1001) = \tfrac{1}{2}$$ and $$Q(0) = \tfrac{1}{2}, \quad Q(1) = \tfrac{1}{2}, \quad Q(1001) = 0.$$
$$\sum_x \sum_{x'} R(x) R(x') \left|x - x'\right|$$
is $250$ and $0.25$, respectively.