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on Ueber eine Eigenschaft des Gaussschen Fehlergesetzes · 1928-stat-21

The statement

Classify all probability density functions $\varphi:\mathbb{R}\to[0,\infty)$ such that, for every integer $n\geq 1$ and every $l_1,\ldots,l_n\in\mathbb{R}$ , the function

L(x)=\prod_{i=1}^{n}\varphi(x-l_i)

is integrable, has a finite first moment, and has a unique global maximizer $l'$ , with this maximizer equal to the centroid $l''$ of $L$ , where

l''=\frac{\int_{-\infty}^{\infty}xL(x)\,dx}{\int_{-\infty}^{\infty}L(x)\,dx}.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.