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Suppose $X>0$ is a random variable with $\mathbb{E}X=c$ and $\mathbb{E}X^{-1}=c^{-1}$.

By applying Cauchy-Schwarz to $\sqrt{X}$ and $\sqrt{X^{-1}}$ and using the "equality iff linearly dependent" case, we have that $X$ must be constant.

But this seems like far too heavy machinery for what ought to be an obvious statement. Is there a simpler proof? Is there an "equality iff" form of Jensen's inequality?

cfp
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    https://math.stackexchange.com/questions/628386/when-jensens-inequality-is-equality?rq=1 –  Apr 20 '22 at 14:44
  • @d.k.o. Though actually this doesn't take you all the way. If $X^{-1}=aX+b$, then there can be two positive solutions for $X$, so it may not be constant. – cfp Apr 20 '22 at 15:26

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