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 Improved Method for Approximating the Hazard Rate for Convolution Poisson Random Variables

2021 ◽  
Author(s):  
Andrei Volodin ◽  
ALYA AL MUTAIRI
Keyword(s):  
Hazard Rate ◽  
Cumulative Distribution Function ◽  
Random Variables ◽  
Saddlepoint Approximation ◽  
Mass Function ◽  
Rate Function ◽  
Cumulative Distribution ◽  
Probability Mass Function ◽  
Complicated Model ◽  
Probability Mass

In this study, we investigate the performance of the saddlepoint approximation of the probability mass function and the cumulative distribution function for the weighted sum of independent Poisson random variables. The goal is to approximate the hazard rate function for this complicated model. The better performance of this method is shown by numerical simulations and comparison with a performance of other approximation methods.

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.

scholarly journals Modified Recursions for a Class of Compound Distributions

1996 ◽  
Vol 26 (2) ◽  
pp. 213-224 ◽  
Author(s):  
Karl-Heinz Waldmann
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Mass Function ◽  
Cumulative Distribution ◽  
Probability Mass Function ◽  
Initial Value ◽  
Compound Distributions ◽  
Claim Frequency ◽  
Probability Mass ◽  
Stop Loss

AbstractRecursions are derived for a class of compound distributions having a claim frequency distribution of the well known (a,b)-type. The probability mass function on which the recursions are usually based is replaced by the distribution function in order to obtain increasing iterates. A monotone transformation is suggested to avoid an underflow in the initial stages of the iteration. The faster increase of the transformed iterates is diminished by use of a scaling function. Further, an adaptive weighting depending on the initial value and the increase of the iterates is derived. It enables us to manage an arbitrary large portfolio. Some numerical results are displayed demonstrating the efficiency of the different methods. The computation of the stop-loss premiums using these methods are indicated. Finally, related iteration schemes based on the cumulative distribution function are outlined.

Quantification of Low-Cycle Fatigue Life for a Gas Turbine Compressor Vane Carrier Under Varying Operating Conditions

2020 ◽  
Author(s):  
Zixi Han ◽  
Zixian Jiang ◽  
Sophie Ehrt ◽  
Mian Li
Keyword(s):  
Low Cycle Fatigue ◽  
Computational Cost ◽  
Mass Function ◽  
Operating Conditions ◽  
Probability Mass Function ◽  
Assessment Approach ◽  
Probability Mass ◽  
Gas Turbine Compressor ◽  
Operating Cycles ◽  
Operational Data

Abstract The design of a gas turbine compressor vane carrier (CVC) should meet mechanical integrity requirements on, among others, low-cycle fatigue (LCF). The number of cycles to the LCF failure is the result of cyclic mechanical and thermal strain effects caused by operating conditions on the components. The conventional LCF assessment is usually based on the assumption on standard operating cycles — supplemented by the consideration of predefined extreme operations and safety factors to compensate a potential underestimate on the LCF damage caused by multiple reasons such as non-standard operating cycles. However, real operating cycles can vary significantly from those standard ones considered in the conventional methods. The conventional prediction of LCF life can be very different from real cases, due to the included safety margins. This work presents a probabilistic method to estimate the distributions of the LCF life under varying operating conditions using operational fleet data. Finite element analysis (FEA) results indicate that the first ramp-up loading in each cycle and the turning time before hot-restart cycles are two predominant contributors to the LCF damage. A surrogate model of LCF damage has been built with regard to these two features to reduce the computational cost of FEA. Miner’s rule is applied to calculate the accumulated LCF damage on the component and then obtain the LCF life. The proposed LCF assessment approach has two special points. First, a new data processing technique inspired by the cumulative sum (CUSUM) control chart is proposed to identify the first ramp-up period of each cycle from noised operational data. Second, the probability mass function of the LCF life for a CVC is estimated using the sequential convolution of the single-cycle damage distribution obtained from operational data. The result from the proposed method shows that the mean value of the LCF life at a critical location of the CVC is significantly larger than the calculated result from the deterministic assessment, and the LCF lives for different gas turbines of the same class are also very different. Finally, to avoid high computational cost of sequential convolution, a quick approximation approach for the probability mass function of the LCF life is given. With the capability of dealing with varying operating conditions and noises in the operational data, the enhanced LCF assessment approach proposed in this work provides a probabilistic reference both for reliability analysis in CVC design, and for predictive maintenance in after-sales service.

Reliability Concepts

2018 ◽  
pp. 1-22
Keyword(s):  
Hazard Rate ◽  
Failure Process ◽  
Reliability Function ◽  
Rate Function ◽  
Cumulative Distribution ◽  
Hazard Rate Function ◽  
Reliability Engineering ◽  
Probability Models ◽  
Engineering Design Process ◽  
Basic Concepts

The first chapter introduces basic concepts of Reliability and their relationships. Four probability functions—reliability function, cumulative distribution function, probability density function, and hazard rate function—that completely characterize the failure process are defined. Three failure rates—MTBF, MTTF, MTTR—that play important role in reliability engineering design process are explained here. The three patterns of failures, DFR, CFR, and IFR, are discussed with reference to the bathtub curve. Two probability models, Exponential and Weibull, are presented. Series and parallel systems and application areas of reliability are also presented.

ON SEVERAL PROPERTIES OF A CLASS OF PREFERENTIAL ATTACHMENT TREES—PLANE-ORIENTED RECURSIVE TREES

2020 ◽  
pp. 1-19
Author(s):  
Panpan Zhang
Keyword(s):  
Continuous Time ◽  
Preferential Attachment ◽  
Mass Function ◽  
Probability Mass Function ◽  
Zagreb Index ◽  
Exact Probability ◽  
Probability Mass ◽  
Proper Scaling ◽  
Recursive Trees ◽  
Degree Variable

In this paper, several properties of a class of trees presenting preferential attachment phenomenon—plane-oriented recursive trees (PORTs) are uncovered. Specifically, we investigate the degree profile of a PORT by determining the exact probability mass function of the degree of a node with a fixed label. We compute the expectation and the variance of degree variable via a Pólya urn approach. In addition, we study a topological index, Zagreb index, of this class of trees. We calculate the exact first two moments of the Zagreb index (of PORTs) by using recurrence methods. Lastly, we determine the limiting degree distribution in PORTs that grow in continuous time, where the embedding is done in a Poissonization framework. We show that it is exponential after proper scaling.

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