Probability distributions unimodality of finite sample extremes of independent Erlang random variables

Samples of independent identically distributed random non-negative values r ∼ 1 , … , r ∼ N with a finite size N ≥ 2 are studied. It is posed the problem to find the sufficient conditions for their common probability distribution Q ( x ) = Pr { r ∼ j < x } , j = 1 ÷ N which guarantee the unimodality of the probability distributions F N ( + ) ( x ) = Pr { r ∼ + < x } and F N ( − ) ( x ) = Pr { r ∼ − < x } which correspond to the maximum r ∼ + = max { r ∼ j ; j = 1 ÷ N } and to the minimum r ∼ − = max { r ∼ j ; j = 1 ÷ N } of the sample, respectively. It is proved that if the distribution Q is determined by a continuously differentiable Erlang probability density q of an arbitrary order n ∈ N then distributions F N ( ± ) are unimodal.