scholarly journals Formulas derived from moment generating functions and Bernstein polynomials

2019 ◽  
Vol 13 (3) ◽  
pp. 839-848 ◽  
Author(s):  
Buket Simsek
Keyword(s):  
Partial Derivative ◽  
Generating Functions ◽  
Functional Equations ◽  
Bernstein Polynomials ◽  
Random Variable ◽  
Discrete Random Variable ◽  
Derivative Formula ◽  
Moment Generating Functions ◽  
Derivative Formulas

The purpose of this paper is to provide some identities derived by moment generating functions and characteristics functions. By using functional equations of the generating functions for the combinatorial numbers y1 (m,n,?), defined in [12, p. 8, Theorem 1], we obtain some new formulas for moments of discrete random variable that follows binomial (Newton) distribution with an application of the Bernstein polynomials. Finally, we present partial derivative formulas for moment generating functions which involve derivative formula of the Bernstein polynomials.

scholarly journals On Generating Functions for Boole Type Polynomials and Numbers of Higher Order and Their Applications

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 352 ◽  
Author(s):  
Yilmaz Simsek ◽  
Ji So
Keyword(s):  
Partial Derivative ◽  
Generating Functions ◽  
Functional Equations ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Higher Order ◽  
Euler Polynomials ◽  
Derivative Formulas ◽  
Combinatorial Sums

The purpose of this manuscript is to study and investigate generating functions for Boole type polynomials and numbers of higher order. With the help of these generating functions, many properties of Boole type polynomials and numbers are presented. By applications of partial derivative and functional equations for these functions, derivative formulas, recurrence relations and finite combinatorial sums involving the Apostol-Euler polynomials, the Stirling numbers and the Daehee numbers are given.

scholarly journals A note on characteristic function for Bernstein polynomials involving special numbers and polynomials

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 543-549
Author(s):  
Buket Simsek
Keyword(s):  
Characteristic Function ◽  
Binomial Distribution ◽  
Bernstein Polynomials ◽  
Random Variable ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Discrete Random Variable ◽  
Euler Numbers ◽  
Negative Order ◽  
Special Numbers And Polynomials

The aim of this present paper is to establish and study generating function associated with a characteristic function for the Bernstein polynomials. By this function, we derive many identities, relations and formulas relevant to moments of discrete random variable for the Bernstein polynomials (binomial distribution), Bernoulli numbers of negative order, Euler numbers of negative order and the Stirling numbers.

scholarly journals Some new identities and formulas for higher-order combinatorial-type numbers and polynomials

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 551-558
Author(s):  
Irem Kucukoglu
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Stirling Numbers ◽  
Higher Order ◽  
Combinatorial Type ◽  
Derivative Formulas ◽  
Cauchy Numbers ◽  
Special Numbers And Polynomials ◽  
Boole Polynomials ◽  
Changhee Numbers

The main purpose of this paper is to provide various identities and formulas for higherorder combinatorial-type numbers and polynomials with the help of generating functions and their both functional equations and derivative formulas. The results of this paper comprise some special numbers and polynomials such as the Stirling numbers of the first kind, the Cauchy numbers, the Changhee numbers, the Simsek numbers, the Peters poynomials, the Boole polynomials, the Simsek polynomials. Finally, remarks and observations on our results are given.

Bounds for moment generating functions and for extinction probabilities

1966 ◽  
Vol 3 (1) ◽  
pp. 171-178 ◽  
Author(s):  
D. Brook
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Natural Extension ◽  
Random Variable ◽  
Moment Generating Function ◽  
Multivariate Case ◽  
Extinction Probabilities ◽  
Moment Generating Functions

Suppose that we have a non-negative, real valued random variable x, whose distribution is governed by some unknown moment generating function M(t). Suppose further that we are given certain moments of x, then the question to be discussed in this paper is : can we find a sharp upper bounding function for the m.g.f.? It will be shown that this is usually possible both in the single variate case and in its natural extension to the multivariate case.

Bounds for moment generating functions and for extinction probabilities

1966 ◽  
Vol 3 (01) ◽  
pp. 171-178 ◽  
Author(s):  
D. Brook
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Natural Extension ◽  
Random Variable ◽  
Moment Generating Function ◽  
Multivariate Case ◽  
Extinction Probabilities ◽  
Moment Generating Functions

Suppose that we have a non-negative, real valued random variable x, whose distribution is governed by some unknown moment generating function M(t). Suppose further that we are given certain moments of x, then the question to be discussed in this paper is : can we find a sharp upper bounding function for the m.g.f.? It will be shown that this is usually possible both in the single variate case and in its natural extension to the multivariate case.

Identities related to special polynomials and combinatorial numbers

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4833-4844 ◽  
Author(s):  
Eda Yuluklu ◽  
Yilmaz Simsek ◽  
Takao Komatsu
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Euler Numbers ◽  
Special Polynomials ◽  
Central Factorial Numbers

The aim of this paper is to give some new identities and relations related to the some families of special numbers such as the Bernoulli numbers, the Euler numbers, the Stirling numbers of the first and second kinds, the central factorial numbers and also the numbers y1(n,k,?) and y2(n,k,?) which are given Simsek [31]. Our method is related to the functional equations of the generating functions and the fermionic and bosonic p-adic Volkenborn integral on Zp. Finally, we give remarks and comments on our results.

Binomial subsampling

2006 ◽  
Vol 462 (2068) ◽  
pp. 1181-1195 ◽  
Author(s):  
Carsten Wiuf ◽  
Michael P.H Stumpf
Keyword(s):  
Mathematical Biology ◽  
Power Series ◽  
Generating Functions ◽  
Binomial Distribution ◽  
Random Variable ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Family ◽  
Necessary And Sufficient ◽  
Finite Moments

In this paper, we discuss statistical families with the property that if the distribution of a random variable X is in , then so is the distribution of Z ∼Bi( X ,  p ) for 0≤ p ≤1. (Here we take Z ∼Bi( X ,  p ) to mean that given X = x ,  Z is a draw from the binomial distribution Bi( x ,  p ).) It is said that the family is closed under binomial subsampling. We characterize such families in terms of probability generating functions and for families with finite moments of all orders we give a necessary and sufficient condition for the family to be closed under binomial subsampling. The results are illustrated with power series and other examples, and related to examples from mathematical biology. Finally, some issues concerning inference are discussed.

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