Unimodality of probability distributions for sample maxima of independent erlang random variables

Finite samples of independent identically distributed nonnegative random variables $\tilde{r}_1, \ldots, \tilde{r}_N$ are under consideration. It is set the problem about sufficient conditions for their common probability distribution $Q(x)=Pr\{\tilde{r}_j<x\}, j=1\div N$ which guarantee the unimodality of the probability distribution $F_N(x)=Pr\{\tilde{r}<x\}$ of random value $\tilde{r}=\max\{\tilde{r}_j; \, j=1\div N\}$. It is proven that in the case when $Q$ has the continuously differentiable density $q$ that is the Erlang density with an arbitrary order $n \in \mathbb{N}$, the distribution $F_N$ is unimodal.