• Journal of the Korean Operations Research and Management Science Society
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• v.18 no.1
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• pp.71-78
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• 1993

A Baysian approach is proposed is estimating the mission failure rates by criticalities. A mission failure which occurs according to a Poisson process with unknown rate is assumed to be classified as one of the criticality levels with an unknown probability. We employ the Gamma prior for the mission failure rate and the Dirichlet prior for the criticality probabilities. Posterior distributions of the mission rates by criticalities and predictive distributions of the time to failure are derived.

• Journal of the Korean Data and Information Science Society
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• v.15 no.1
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• pp.273-284
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• 2004

Bayes estimation of parameters is considered for two independent exponential distributions with ordered means. Order restricted Bayes estimators for means are obtained with respect to inverted gamma, noninformative prior and uniform prior distributions, and their asymptotic properties are established. It is shown that the maximum likelihood estimator, restricted maximum likelihood estimator, unrestricted Bayes estimator, and restricted Bayes estimator of the mean are all consistent and have the same limiting distribution. These estimators are compared with the corresponding unrestricted Bayes estimators by Monte Carlo simulation.

• Journal of Applied Reliability
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• v.11 no.4
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• pp.319-330
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• 2011

We consider the stress-strength model in which a unit of strength $T_2$ is subjected to environmental stress $T_1$. An important measure considered in stress-strength model is the reliability parameter R=P($T_2$ > $T_1$). The greater the value of R is, the more reliable is the unit to perform its specified task. In this article, we consider the situations in which $T_1$ and $T_2$ are both independent and dependent, and have certain bivariate distributions as their joint distributions. To study the effect of dependency on R, we investigate several bivariate distributions of $T_1$ and $T_2$ and compare the values of R for these distributions. Numerical comparisons are presented depending on the parameter values as well.

• Journal of the Chungcheong Mathematical Society
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• v.28 no.2
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• pp.187-194
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• 2015

The concept of sub-independence is based on the convolution of the distributions of the random variables. It is much weaker than that of independence, but is shown to be sufficient to yield the conclusions of important theorems and results in probability and statistics. It also provides a measure of dissociation between two random variables which is much stronger than uncorrelatedness. Inspired by the excellent work of Jin and Lee (2014), we present certain characterizations of gamma distribution based on the concept of sub-independence.

• Journal of Electrical Engineering and information Science
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• v.2 no.6
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• pp.123-130
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• 1997

This paper presents a generalized expansion method for calculating reliability index in power generation systems. This generalized expansion with a gamma distribution is a very useful tool for the approximation of capacity outage probability distribution of generation system. The well-known Gram-Charlier expansion and Legendre series are also studied in this paper to be compared with this generalized expansion using a sample system IEEE-RTS(Reliability Test System). The results show that the generalized expansion with a composite of gamma distributions is more accurate and stable than Gram-Charlier expansion and Legendre series as addition of the terms to be expanded.

• Bulletin of the Korean Mathematical Society
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• v.56 no.1
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• pp.131-150
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• 2019

In this paper, we consider the structure of ${\gamma}$-constacyclic codes of length $2p^s$ over the Galois ring $GR(p^a,m)$ for any unit ${\gamma}$ of the form ${\xi}_0+p{\xi}_1+p^2z$, where $z{\in}GR(p^a,m)$ and ${\xi}_0$, ${\xi}_1$ are nonzero elements of the set ${\mathcal{T}}(p,m)$. Here ${\mathcal{T}}(p,m)$ denotes a complete set of representatives of the cosets ${\frac{GR(p^a,m)}{pGR(p^a,m)}}={\mathbb{F}}p^m$ in $GR(p^a,m)$. When ${\gamma}$ is not a square, the rings ${\mathcal{R}}_p(a,m,{\gamma})=\frac{GR(p^a,m)[x]}{{\langle}x^2p^s-{\gamma}{\rangle}}$ is a chain ring with maximal ideal ${\langle}x^2-{\delta}{\rangle}$, where ${\delta}p^s={\xi}_0$, and the number of codewords of ${\gamma}$-constacyclic code are provided. Furthermore, the self-orthogonal and self-dual ${\gamma}$-constacyclic codes of length $2p^s$ over $GR(p^a,m)$ are also established. Finally, we determine the Rosenbloom-Tsfasman (RT) distances and weight distributions of all such codes.

• Proceedings of the Korean Society for Quality Management Conference
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• 1998.11a
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• pp.133-142
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• 1998

This paper is a brief review of the different procedures that are available for fitting theoretical distributions to data. The use of each technique is illustrated by reference to a distribution system which including the Pearson, Poission approximation of Gamma distribution and Burr functions. These functions can be used to calculate percent out of specification. Therefore, in this paper a new methods for estimating a measure of non-normal process capability for Gamma distributed variable data proposed using the percentage nonconforming. Process capability indices combines with the percentage nonconforming information can be used to evaluate more accurately process capability.

• Journal of the Chungcheong Mathematical Society
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• v.27 no.2
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• pp.157-163
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• 2014

Let {$X_i$, $1{\leq}i{\leq}n$} be a sequence of i.i.d. sequence of positive random variables with common absolutely continuous cumulative distribution function F(x) and probability density function f(x) and $E(X^2)$ < ${\infty}$. The random variables X + Y and $\frac{(X-Y)^2}{(X+Y)^2}$ are independent if and only if X and Y have gamma distributions. In addition, the random variables $S_n$ and $\frac{\sum_{i=1}^{m}(X_i)^2}{(S_n)^2}$ with $S_n=\sum_{i=1}^{n}X_i$ are independent for $1{\leq}m$ < n if and only if $X_i$ has gamma distribution for $i=1,{\cdots},n$.

• International Journal of Reliability and Applications
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• v.16 no.2
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• pp.81-98
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• 2015

The Exponentiated Gamma (EG) distribution is one of the important families of distributions in lifetime tests. In this paper, a new generalized version of this distribution which is called kumaraswamy Exponentiated Gamma (KEG) distribution is introduced. A new distribution is more flexible and has some interesting properties. A comprehensive mathematical treatment of the KEG distribution is provided. We derive the $r^{th}$ moment and moment generating function of this distribution. Moreover, we discuss the maximum likelihood estimation of the distribution parameters. Finally, an application to real data sets is illustrated.

• Progress in Medical Physics
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• v.28 no.4
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• pp.207-217
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• 2017

For effective patient treatment in proton therapy, it is therefore important to accurately measure the beam range. For measuring beam range, various researchers determine the beam range by measuring the prompt gammas generated during nuclear reactions of protons with materials. However, the accuracy of the beam range determination can be lowered in heterogeneous phantoms, because of the differences with respect to the prompt gamma production depending on the properties of the material. In this research, to improve the beam range determination in a heterogeneous phantom, we derived a formula to correct the prompt-gamma distribution using the ratio of the prompt gamma production, stopping power, and density obtained for each material. Then, the prompt-gamma distributions were acquired by a multi-slit prompt-gamma camera on various kinds of heterogeneous phantoms using a Geant4 Monte Carlo simulation, and the deduced formula was applied to the prompt-gamma distributions. For the case involving the phantom having bone-equivalent material in the soft tissue-equivalent material, it was confirmed that compared to the actual range, the determined ranges were relatively accurate both before and after correction. In the case of a phantom having the lung-equivalent material in the soft tissue-equivalent material, although the maximum error before correction was 18.7 mm, the difference was very large. However, when the correction method was applied, the accuracy was significantly improved by a maximum error of 4.1 mm. Moreover, for a phantom that was constructed based on CT data, after applying the calibration method, the beam range could be generally determined within an error of 2.5 mm. Simulation results confirmed the potential to determine the beam range with high accuracy in heterogeneous phantoms by applying the proposed correction method. In future, these methods will be verified by performing experiments using a therapeutic proton beam.