The medians for exponential families and the normal law

Gerard Letac, Mauro Piccioni

Published 2017-08-12Version 1

Let $P$ a probability on the real line generating a natural exponential family $(P_t)_{t\in \R}$. The property that $t$ is a median of $P_t$ for all $t$ is fulfilled by the standard Gaussian law $N(0.1).$ We show that this property is stable in the sense that if $P$ has a bounded density with respect to $N(0,1)$ then $P=N(0,1).$