Abstract
Samples of independent identically distributed random non-negative values with a finite size are studied. It is posed the problem to find the sufficient conditions for their common probability distribution which guarantee the unimodality of the probability distributions and which correspond to the maximum and to the minimum 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 then distributions are unimodal.
Highlights
Historically, the theory of probability distributions of extremes of random variables samples and the theory of the probability distributions of extreme values of trajectories of stationary random processes which is connected with it have arisen on the basis of the necessity to find some answers of quite practical questions
This theory has extracted into a separate direction of research in the probability theory
The statistical characteristics of extremes of independent random variables were studied in these works, where some probability distributions were chosen for them which were already widely used in probability theory
Summary
The theory of probability distributions of extremes of random variables samples and the theory of the probability distributions of extreme values of trajectories of stationary random processes which is connected with it have arisen on the basis of the necessity to find some answers of quite practical questions. The statistical characteristics of extremes of independent random variables were studied in these works, where some probability distributions were chosen for them which were already widely used in probability theory. In contrary to such investigations, in the works of M.
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