A probabilistic interpretation of the Bell polynomials

2021 ◽  
pp. 1-13
Author(s):  
Kuldeep Kumar Kataria ◽  
Palaniappan Vellaisamy ◽  
Vijay Kumar
Keyword(s):  
Bell Polynomials ◽  
Probabilistic Interpretation

Using Bayesian network models to incorporate uncertainty in the economic analysis of pollution abatement measures under the water framework directive

2005 ◽  
Vol 5 (6) ◽  
pp. 95-104 ◽  
Author(s):  
D.N. Barton ◽  
T. Saloranta ◽  
T.H. Bakken ◽  
A. Lyche Solheim ◽  
J. Moe ◽  
...  
Keyword(s):  
Water Quality ◽  
At Risk ◽  
Economic Valuation ◽  
Pollution Abatement ◽  
Network Models ◽  
Cost Effective ◽  
Bayesian Belief Networks ◽  
Probabilistic Interpretation ◽  
Belief Networks ◽  
Benefit Cost Analysis

The evaluation of water bodies “at risk” of not achieving the Water Framework Directive's (WFD) goal of “good status” begs the question of how big a risk is acceptable before a programme of measures should be implemented. Documentation of expert judgement and statistical uncertainty in pollution budgets and water quality modelling, combined with Monte Carlo simulation and Bayesian belief networks, make it possible to give a probabilistic interpretation of “at risk”. Combined with information on abatement costs, a cost-effective ranking of measures based on expected costs and effect can be undertaken. Combined with economic valuation of water quality, the definition of “disproportionate cost” of abatement measures compared to benefits of achieving “good status” can also be given a probabilistic interpretation. Explicit modelling of uncertainty helps visualize where research and consulting efforts are most critical for reducing uncertainty. Based on data from the Morsa catchment in South-Eastern Norway, this paper discusses the relative merits of using Bayesian belief networks when integrating biophysical modelling results in the benefit-cost analysis of derogations and cost-effectiveness ranking of abatement measures under the WFD.

Asymptotic behaviour of sample weighted circuits representing recurrent Markov chains

1990 ◽  
Vol 27 (03) ◽  
pp. 545-556 ◽  
Author(s):  
S. Kalpazidou
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Asymptotic Behaviour ◽  
Probabilistic Interpretation ◽  
Stationary Markov Chain

The asymptotic behaviour of the sequence (𝒞 n (ω), wc,n (ω)/n), is studied where 𝒞 n (ω) is the class of all cycles c occurring along the trajectory ωof a recurrent strictly stationary Markov chain (ξ n ) until time n and wc,n (ω) is the number of occurrences of the cycle c until time n. The previous sequence of sample weighted classes converges almost surely to a class of directed weighted cycles (𝒞∞, ω c ) which represents uniquely the chain (ξ n ) as a circuit chain, and ω c is given a probabilistic interpretation.

scholarly journals Explicit, Determinantal, and Recurrent Formulas of Generalized Eulerian Polynomials

Axioms ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 37
Author(s):  
Yan Wang ◽  
Muhammet Cihat Dağli ◽  
Xi-Min Liu ◽  
Feng Qi
Keyword(s):  
General Formula ◽  
Bell Polynomials ◽  
Eulerian Polynomials ◽  
Differentiable Functions ◽  
Faà Di Bruno Formula ◽  
Derivatives Of

In the paper, by virtue of the Faà di Bruno formula, with the aid of some properties of the Bell polynomials of the second kind, and by means of a general formula for derivatives of the ratio between two differentiable functions, the authors establish explicit, determinantal, and recurrent formulas for generalized Eulerian polynomials.

scholarly journals Some identities for degenerate complete and incomplete r-Bell polynomials

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Jongkyum Kwon ◽  
Taekyun Kim ◽  
Dae San Kim ◽  
Han Young Kim
Keyword(s):  
Bell Polynomials

On a generalization of the Rényi–Srivastava characterization of the Poisson law

2021 ◽  
Vol 58 (1) ◽  
pp. 68-82
Author(s):  
Jean-Renaud Pycke
Keyword(s):  
Life Insurance ◽  
Collective Model ◽  
New Method ◽  
Bell Polynomials ◽  
Explicit Formulas ◽  
Compound Distributions ◽  
Poisson Law ◽  
Pierre Loti

AbstractWe give a new method of proof for a result of D. Pierre-Loti-Viaud and P. Boulongne which can be seen as a generalization of a characterization of Poisson law due to Rényi and Srivastava. We also provide explicit formulas, in terms of Bell polynomials, for the moments of the compound distributions occurring in the extended collective model in non-life insurance.

scholarly journals Fully degenerate Bell polynomials associated with degenerate Poisson random variables

2021 ◽  
Vol 19 (1) ◽  
pp. 284-296
Author(s):  
Hye Kyung Kim
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Combinatorial Identities ◽  
Bell Polynomials ◽  
Poisson Random Variable ◽  
Central Moments ◽  
Special Polynomials ◽  
Special Numbers And Polynomials ◽  
New Type ◽  
Two Parameters

Abstract Many mathematicians have studied degenerate versions of quite a few special polynomials and numbers since Carlitz’s work (Utilitas Math. 15 (1979), 51–88). Recently, Kim et al. studied the degenerate gamma random variables, discrete degenerate random variables and two-variable degenerate Bell polynomials associated with Poisson degenerate central moments, etc. This paper is divided into two parts. In the first part, we introduce a new type of degenerate Bell polynomials associated with degenerate Poisson random variables with parameter α > 0 \alpha \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the fully degenerate Bell polynomials. We derive some combinatorial identities for the fully degenerate Bell polynomials related to the n n th moment of the degenerate Poisson random variable, special numbers and polynomials. In the second part, we consider the fully degenerate Bell polynomials associated with degenerate Poisson random variables with two parameters α > 0 \alpha \gt 0 and β > 0 \beta \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the two-variable fully degenerate Bell polynomials. We show their connection with the degenerate Poisson central moments, special numbers and polynomials.

scholarly journals The r-Dowling–Lah Polynomials

2021 ◽  
Vol 18 (4) ◽  
Author(s):  
Eszter Gyimesi
Keyword(s):  
Bell Polynomials ◽  
Close Relatives ◽  
Comprehensive Study

AbstractThe notions of r-Bell polynomials and their generalization, the r-Dowling polynomials are due to Mező and Cheon, Jung. Recently, Nyul and Rácz defined the r-Lah polynomials, which are close relatives of r-Bell polynomials. In the present paper, we introduce the Dowling type generalization of r-Lah polynomials, the r-Dowling–Lah polynomials. We give a comprehensive study of them using the results of the author and Nyul on r-Whitney–Lah numbers, which are the coefficients of these polynomials.

scholarly journals Complete and incomplete Bell polynomials associated with Lah–Bell numbers and polynomials

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Lee-Chae Jang ◽  
Hyunseok Lee ◽  
Han-Young Kim
Keyword(s):  
Bell Polynomials ◽  
Bell Numbers ◽  
Bell Number ◽  
Finite Sums

AbstractThe nth r-extended Lah–Bell number is defined as the number of ways a set with $n+r$ n + r elements can be partitioned into ordered blocks such that r distinguished elements have to be in distinct ordered blocks. The aim of this paper is to introduce incomplete r-extended Lah–Bell polynomials and complete r-extended Lah–Bell polynomials respectively as multivariate versions of r-Lah numbers and the r-extended Lah–Bell numbers and to investigate some properties and identities for these polynomials. From these investigations we obtain some expressions for the r-Lah numbers and the r-extended Lah–Bell numbers as finite sums.

scholarly journals Degenerate poly-Bell polynomials and numbers

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Taekyun Kim ◽  
Hye Kyung Kim
Keyword(s):  
Combinatorial Identities ◽  
Bell Polynomials ◽  
Genocchi Polynomials ◽  
Special Polynomials ◽  
Explicit Representations

AbstractNumerous mathematicians have studied ‘poly’ as one of the generalizations to special polynomials, such as Bernoulli, Euler, Cauchy, and Genocchi polynomials. In relation to this, in this paper, we introduce the degenerate poly-Bell polynomials emanating from the degenerate polyexponential functions which are called the poly-Bell polynomials when $\lambda \rightarrow 0$ λ → 0 . Specifically, we demonstrate that they are reduced to the degenerate Bell polynomials if $k = 1$ k = 1 . We also provide explicit representations and combinatorial identities for these polynomials, including Dobinski-like formulas, recurrence relationships, etc.

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