stylistic:style_transfer, information:classification
Characteristic property

The second moment of a random variable attains the minimum value when taken around the first moment (i.e., mean) of the random variable, i.e. a r g m i n m E ((X − m) 2) = E (X) {\displaystyle \mathrm {argmin} _{m}\,\mathrm {E} \left(\left(X-m\right)^{2}\right)=\mathrm {E} (X)}. Conversely, if a continuous function φ {\displaystyle \varphi } satisfies a r g m i n m E (φ (X − m)) = E (X) {\displaystyle \mathrm {argmin} _{m}\,\mathrm {E} (\varphi (X-m))=\mathrm {E} (X)} for all random variables X, then it is necessarily of the form φ (x) = a x 2 + b {\displaystyle \varphi (x)=ax^{2}+b}, where a > 0. This also holds in the multidimensional case.