  Let $P_0$ be a probability on the real line generating a natural exponential
family $(P_t)_{t\in \mathbb {R}}$. Fix $\alpha$ in $ (0,1).$ We show that the
property that $P_t((-\infty,t)) \leq \alpha \leq P_t((-\infty,t])$ for all $t$
implies that there exists a number $\mu_\alpha$ such that $P_0$ is the Gaussian
distribution $N(\mu_{\alpha},1).$ In other terms, if for all $t$, $t$ is a
quantile of $P_t$ associated to some threshold $\alpha\in (0,1)$, then the
exponential family must be Gaussian. The case $\alpha=1/2$, \textit{i.e.} $t$
is always a median of $P_t,$ has been considered in Letac \textit{et al.}
(2018). Analogously let $Q$ be a measure on $[0,\infty)$ generating a natural
exponential family $(Q_{-t})_{t>0}$. We show that $Q_{-t}([0,t^{-1}))\leq
\alpha \leq Q_{-t}([0,t^{-1}])$ for all $t>0$ implies that there exists a
number $p=p_{\alpha}>0$ such that $Q(dx)\propto x^{p-1}dx,$ and thus $Q_{-t}$
has to be a gamma distribution with parameters $p$ and $t.$
