Abstract
The main aim of this paper is to define and investigate a new class of symmetric beta type distributions with the help of the symmetric Bernstein-type basis functions. We give symmetry property of these distributions and the Bernstein-type basis functions. Using the Bernstein-type basis functions and binomial series, we give some series and integral representations including moment generating function for these distributions. Using generating functions and their functional equations, we also give many new identities related to the moments, the polygamma function, the digamma function, the harmonic numbers, the Stirling numbers, generalized harmonic numbers, the Lah numbers, the Bernstein-type basis functions, the array polynomials, and the Apostol–Bernoulli polynomials. Moreover, some numerical values of the expected values for the logarithm of random variable are given.
Highlights
Symmetric probability distribution, symmetric random variables, and the symmetric product of two independent random variables have many applications in pure and applied mathematics.The symmetric beta distribution has very important applications in scientific studies
Some mathematical models related to the behavior of random variables, limited to intervals of finite length have been studied
In this paper, using the Bernstein-type basis functions of order n, we defined a new class of beta type distributions
Summary
Symmetric random variables, and the symmetric product of two independent random variables have many applications in pure and applied mathematics. Some mathematical models related to the behavior of random variables, limited to intervals of finite length have been studied. For this reason, this distribution and its applications have been used in mathematics, in probability, in theoretical statistics, in science, in health science, and in social sciences (cf [1,2,3,4,5] and the references cited therein). We give numerical values of the expected value for the logarithm of random variable of the beta type distribution These values are given in terms of the polygamma function, the digamma function. Γ(z) dz (cf. [3,24,37,38] and the references cited therein)
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