Does this property hold for all unimodal functions

24.03.2020, 11:20	recallme	Auf diesen Beitrag antworten »
Does this property hold for all unimodal functions This is the property I would like to prove: $\begin{eqnarray} -G(x)x + B = \int_x^B G(k) dk \end{eqnarray}$ has at most one solution $\begin{eqnarray} x\in [A,B] \end{eqnarray}$ for all unimodal distributions $\begin{eqnarray} g(x) \end{eqnarray}$ (with c.d.f. $\begin{eqnarray} G(x) \end{eqnarray}$ ), with support $\begin{eqnarray} [A,B] \end{eqnarray}$ and $\begin{eqnarray} B> A\geq 0 \end{eqnarray}$ . I was able to prove this property for the uniform distribution, but I struggle to do so for the general case. Perhaps there is also a geometric proof which I am just unable to see at the moment. Any help is highly appreciated Andreas

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