On a characterization of probability distributions by conditional expectations. (English) Zbl 0666.62009
Let X be a random variable with a differentiable disribution function F, let (a,b) denote the unique interval such that \(0<F(x)<1\) for \(a<x<b\) and \(F(x)=1\) or 0 for \(x\not\in (a,b)\), and, for \(a<x<b\), define
\[
K_{\alpha}(x)=E(F^{-\alpha}(X)| X<x).
\]
The author shows that the function \(K_{\alpha}\) uniquely determines the distribution of X via the formula
\[
F(x)=(l-\alpha)^{\alpha}K^{\alpha}_{\alpha}(x),\quad a<x<b.
\]
Reviewer: B.K.Horkelheimer
MSC:
62E10 | Characterization and structure theory of statistical distributions |