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Q: Showing a strictly function has unique solution.

UnknownTheorem: If $X$ has an absolutely continuous distribution $F(t)$ increasing residual entropy $H(f;t)$ , then $H(f;t)$ has uniquely determined $F(t)$. Partial Proof: By definition $$H(f;t)=\int_t^\infty \frac{f(t)}{(1-F(t))}\log\frac{f(t)}{(1-F(t))} $$ which is equivalent to $$f(t)\log f(t)=f(t)(1...

 

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