This article is cited in 1 paper

Unimodality of the probability distribution of the extensive functional of samples of a random sequence

Yu. P. Virchenko^a, A. M. Tevolde<sup>b

a</sup> Belgorod State Technological University named after V. G. Shukhov, Belgorod, Russia
^b Belgorod State University, Belgorod, Russia

Abstract: We establish a criterion for the unimodality of the probability distribution of a functional that is represented by the sum of a set of independent identically distributed random nonnegative variables ${\tilde x}_k$ with a random number of terms distributed according to Poisson. The general distribution of terms ${\tilde x}_k$ is concentrated on the interval $[0, 1]$ and is such that $\mathrm{Pr}\,\{{\tilde x}_k = 0\} \ne 0.$ Its absolutely continuous part is asymptotically close to a uniform distribution. We introduce the concept of smoothing functions and establish an explicit form of the distribution of any fixed number of terms uniformly distributed on $[0, 1].$

Keywords: sum of independent identically distributed random variables, unimodality of probability distribution, smoothing function, single-peak function.

UDC: 519.213

DOI: 10.22363/2413-3639-2024-70-4-542-560