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Theorem: 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...
