Weibull Distributed Stress-Dependent Strength Analysis of Aeroengine Alloy Using Lagrange Factor Polynomial

2010 ◽  
Author(s):  
Chun Nam Wong ◽  
Hong-Zhong Huang ◽  
Jingqi Xiong ◽  
Tianyou Hu
Keyword(s):  
Random Variable ◽  
Mass Function ◽  
Strength Distribution ◽  
Interference Model ◽  
Strength Analysis ◽  
Probability Mass Function ◽  
Application Range ◽  
Probability Mass ◽  
Stress Dependent ◽  
Mean Function

In this paper, the unilateral dependency of strength on stress is taken into account. And the stress-dependent strength is represented by a discrete random variable that has different conditional probability mass functions under different stress amplitudes. Then the Lagrange factor polynomial technique is developed to generate the stress-strength interference model with stress-dependent strength. This model assumes that the strength probability mass function is Weibull distributed, while the stress probability mass function is Normal distributed. Accuracy of this method is investigated by an aeroengine bearing cage alloy. Structural reliabilities are computed as 0.796 to 0.986 under several operation modes, which are analyzed by varying the Weibull shape parameter from 1 to 6. Then probability mean function estimated by Lagrange factor polynomial has relatively low errors over most span of the stress dependent strength distribution. With this approach stress-dependent strength reliability of aeroengine structural systems can be established conveniently. Meanwhile the application range of the classical stress-strength interference model can be extended.

Part Reliability Design of Air-Cooled Turbine Blade Under Thermal Stress-Dependent Strength Using Polynomial Method

2010 ◽  
Author(s):  
Chun Nam Wong ◽  
Hong-Zhong Huang ◽  
Jingqi Xiong ◽  
Daijun Ling
Keyword(s):  
Thermal Stress ◽  
Turbine Blade ◽  
Strength Distribution ◽  
Interference Model ◽  
Polynomial Method ◽  
Structural Systems ◽  
Reliability Design ◽  
Application Range ◽  
Stress Dependent ◽  
Mean Function

In this paper, Lagrange factor polynomial method is developed to generate the stress-strength interference model with thermal stress-dependent strength. Accuracy of this method is investigated by an aeroengine air-cooled turbine blade. The computed reliability is quite high under several thermal stress modes. Then probability mean function estimated by this method has relatively low errors over most subintervals of the thermal stress dependent strength distribution. With this approach conditional thermal stress-dependent strength reliability of aeroengine structural systems can be established conveniently. Meanwhile the application range of the classical stress-strength interference model can be extended.

scholarly journals λ-Analogues of Stirling polynomials of the first kind and their applications

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Han Young Kim ◽  
Sung-Soo Pyo
Keyword(s):  
Recurrence Relations ◽  
Random Variable ◽  
Stirling Numbers ◽  
Mass Function ◽  
Probability Mass Function ◽  
Discrete Random Variable ◽  
Special Polynomials ◽  
Binomial Distributions ◽  
Natural Extensions ◽  
Probability Mass

AbstractRecently, λ-analogues of Stirling numbers of the first kind were studied. In this paper, we introduce, as natural extensions of these numbers, λ-Stirling polynomials of the first kind and r-truncated λ-Stirling polynomials of the first kind. We give recurrence relations, explicit expressions, some identities, and connections with other special polynomials for those polynomials. Further, as applications, we show that both of them appear in an expression of the probability mass function of a suitable discrete random variable, constructed from λ-logarithmic and negative λ-binomial distributions.

scholarly journals A Probability Mass Function for Various Shapes of the Failure Rates, Asymmetric and Dispersed Data with Applications to Coronavirus and Kidney Dysmorphogenesis

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1790
Author(s):  
Mahmoud El-Morshedy ◽  
Morad Alizadeh ◽  
Afrah Al-Bossly ◽  
Mohamed S. Eliwa
Keyword(s):  
Hazard Rate ◽  
Geometric Model ◽  
Mass Function ◽  
Rate Function ◽  
Discrete Analogue ◽  
Strength Analysis ◽  
Residual Entropy ◽  
Probability Mass Function ◽  
Hazard Rate Function ◽  
Probability Mass

In this article, a discrete analogue of an extension to a two-parameter half-logistic model is proposed for modeling count data. The probability mass function of the new model can be expressed as a mixture representation of a geometric model. Some of its statistical properties, including hazard rate function, moments, moment generating function, conditional moments, stress-strength analysis, residual entropy, cumulative residual entropy and order statistics with its moments, are derived. It is found that the new distribution can be utilized to model positive skewed data, and it can be used for analyzing equi- and over-dispersed data. Furthermore, the hazard rate function can be either decreasing, increasing or bathtub. The parameter estimation through the classical point of view has been performed using the method of maximum likelihood. A detailed simulation study is carried out to examine the outcomes of the estimators. Finally, two distinctive real data sets are analyzed to prove the flexibility of the proposed discrete distribution.

scholarly journals On Some Compound Random Variables Motivated by Bulk Queues

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Romeo Meštrović
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Mass Function ◽  
Probability Mass Function ◽  
Time Of Arrival ◽  
Uniform Random Variable ◽  
Wide Range ◽  
Bulk Queue ◽  
Probability Mass ◽  
Number Of Customers

We consider the distribution of the number of customers that arrive in an arbitrary bulk arrival queue system. Under certain conditions on the distributions of the time of arrival of an arriving group (Y(t)) and its size (X) with respect to the considered bulk queue, we derive a general expression for the probability mass function of the random variableQ(t)which expresses the number of customers that arrive in this bulk queue during any considered periodt. Notice thatQ(t)can be considered as a well-known compound random variable. Using this expression, without the use of generating function, we establish the expressions for probability mass function for some compound distributionsQ(t)concerning certain pairs(Y(t),X)of discrete random variables which play an important role in application of batch arrival queues which have a wide range of applications in different forms of transportation. In particular, we consider the cases whenY(t)and/orXare some of the following distributions: Poisson, shifted-Poisson, geometrical, or uniform random variable.

Evaluating Best-Case and Worst-Case Coefficients of Variation when Bounds Are Available

1992 ◽  
Vol 6 (3) ◽  
pp. 309-322 ◽  
Author(s):  
George S. Fishman ◽  
David S. Rubin
Keyword(s):  
Network Reliability ◽  
Random Variable ◽  
Mass Function ◽  
Probability Mass Function ◽  
Lower And Upper Bounds ◽  
Worst Case ◽  
The Monte Carlo Method ◽  
Coefficients Of Variation ◽  
Probability Mass ◽  
Bounded Random Variable

This paper describes a procedure for computing tightest possible best-case and worst-case bounds on the coefficient of variation of a discrete, bounded random variable when lower and upper bounds are available for its unknown probability mass function. An example from the application of the Monte Carlo method to the estimation of network reliability illustrates the procedure and, in particular, reveals considerable tightening in the worst-case bound when compared to the trivial worst-case bound based exclusively on range.

The Back Page: My Favorite Lesson: Perfect Sample and Population Mean Formula

2013 ◽  
Vol 107 (4) ◽  
pp. 320
Author(s):  
Clarence W. Lienhard
Keyword(s):  
Random Variable ◽  
Mass Function ◽  
Probability Mass Function ◽  
Discrete Random Variable ◽  
Sample Mean ◽  
Population Mean ◽  
Probability Mass

Typically, students readily understand the concepts of the sample mean and a discrete random variable X with its probability mass function. However, they find the population mean formula unduly difficult to comprehend. Using a perfect sample, the author guides readers to discover the population mean formula from the sample mean.

Maximum Likelihood Set for Estimating a Probability Mass Function

2005 ◽  
Vol 17 (7) ◽  
pp. 1508-1530 ◽  
Author(s):  
Bruno M. Jedynak ◽  
Sanjeev Khudanpur
Keyword(s):  
Maximum Likelihood ◽  
Prior Knowledge ◽  
Empirical Distribution ◽  
Random Variable ◽  
Mass Function ◽  
Small Sample ◽  
Probability Mass Function ◽  
Probability Mass ◽  
Dirichlet Prior ◽  
Probability Simplex

We propose a new method for estimating the probability mass function (pmf) of a discrete and finite random variable from a small sample. We focus on the observed counts—the number of times each value appears in the sample—and define the maximum likelihood set (MLS) as the set of pmfs that put more mass on the observed counts than on any other set of counts possible for the same sample size. We characterize the MLS in detail in this article. We show that the MLS is a diamond-shaped subset of the probability simplex [0, 1]k bounded by at most k × (k − 1) hyper-planes, where k is the number of possible values of the random variable. The MLS always contains the empirical distribution, as well as a family of Bayesian estimators based on a Dirichlet prior, particularly the well-known Laplace estimator. We propose to select from the MLS the pmf that is closest to a fixed pmf that encodes prior knowledge. When using Kullback-Leibler distance for this selection, the optimization problem comprises finding the minimum of a convex function over a domain defined by linear inequalities, for which standard numerical procedures are available. We apply this estimate to language modeling using Zipf's law to encode prior knowledge and show that this method permits obtaining state-of-the-art results while being conceptually simpler than most competing methods.

A study on the injective coloring parameters of certain graphs

2016 ◽  
Vol 08 (03) ◽  
pp. 1650052 ◽  
Author(s):  
N. K. Sudev ◽  
K. P. Chithra ◽  
S. Satheesh ◽  
Johan Kok
Keyword(s):  
Graph Coloring ◽  
Random Variable ◽  
Mass Function ◽  
Probability Mass Function ◽  
Proper Coloring ◽  
Discrete Random Variable ◽  
Injective Coloring ◽  
Random Experiment ◽  
Mean And Variance ◽  
Probability Mass

Coloring the vertices of a graph [Formula: see text] according to certain conditions can be considered as a random experiment and a discrete random variable (r.v.) [Formula: see text] can be defined as the number of vertices having a particular color in the proper coloring of [Formula: see text] and a probability mass function for this random variable can be defined accordingly. In this paper, we extend the concepts of mean and variance to a modified injective graph coloring and determine the values of these parameters for a number of standard graphs.

scholarly journals Discrete uniform and binomial distributions with infinite support

2020 ◽  
Vol 24 (23) ◽  
pp. 17517-17524 ◽  
Author(s):  
Andrey Pepelyshev ◽  
Anatoly Zhigljavsky
Keyword(s):  
Binomial Distribution ◽  
Numerical Study ◽  
Probability Distributions ◽  
Random Variable ◽  
Mass Function ◽  
Probability Mass Function ◽  
Binomial Distributions ◽  
Probability Mass ◽  
Discrete Uniform ◽  
Infinite Set

AbstractWe study properties of two probability distributions defined on the infinite set $$\{0,1,2, \ldots \}$$ { 0 , 1 , 2 , … } and generalizing the ordinary discrete uniform and binomial distributions. Both extensions use the grossone-model of infinity. The first of the two distributions we study is uniform and assigns masses $$1/\textcircled {1}$$ 1 / 1 to all points in the set $$ \{0,1,\ldots ,\textcircled {1}-1\}$$ { 0 , 1 , … , 1 - 1 } , where $$\textcircled {1}$$ 1 denotes the grossone. For this distribution, we study the problem of decomposing a random variable $$\xi $$ ξ with this distribution as a sum $$\xi {\mathop {=}\limits ^\mathrm{d}} \xi _1 + \cdots + \xi _m$$ ξ = d ξ 1 + ⋯ + ξ m , where $$\xi _1 , \ldots , \xi _m$$ ξ 1 , … , ξ m are independent non-degenerate random variables. Then, we develop an approximation for the probability mass function of the binomial distribution Bin$$(\textcircled {1},p)$$ ( 1 , p ) with $$p=c/\textcircled {1}^{\alpha }$$ p = c / 1 α with $$1/2<\alpha \le 1$$ 1 / 2 < α ≤ 1 . The accuracy of this approximation is assessed using a numerical study.

On certain parameters of equitable coloring of graphs

2017 ◽  
Vol 09 (04) ◽  
pp. 1750054 ◽  
Author(s):  
N. K. Sudev ◽  
K. P. Chithra ◽  
S. Satheesh ◽  
Johan Kok
Keyword(s):  
Random Variable ◽  
Mass Function ◽  
Probability Mass Function ◽  
Proper Coloring ◽  
Discrete Random Variable ◽  
Statistical Parameters ◽  
Equitable Coloring ◽  
Mean And Variance ◽  
Probability Mass ◽  
The Given

Coloring the vertices of a graph [Formula: see text] according to certain conditions is a random experiment and a discrete random variable [Formula: see text] is defined as the number of vertices having a particular color in the given type of coloring of [Formula: see text] and a probability mass function for this random variable can be defined accordingly. An equitable coloring of a graph [Formula: see text] is a proper coloring [Formula: see text] of [Formula: see text] which an assignment of colors to the vertices of [Formula: see text] such that the numbers of vertices in any two color classes differ by at most one. In this paper, we extend the concepts of arithmetic mean and variance, the two major statistical parameters, to the theory of equitable graph coloring and hence determine the values of these parameters for a number of standard graphs.

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