• Communications for Statistical Applications and Methods
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• v.12 no.2
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• pp.435-442
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• 2005

The mathematical properties of the sequentially operated systems are often described by integral equations. Reservoir system of a product and sequential probability ratio test (SPRT) are typical examples of sequentially operated systems. When the underlying random quantities follow Erlang distribution, a systematic method was developed to solve the integral equations. We extend the method to the cases having accrual functions of more general types. The solutions of the integral equations are represented as a linear combination of distribution functions, and the coefficients of the linear combination are obtained by solving linear system derived from the continuity condition of the solutions.

• Bulletin of the Korean Mathematical Society
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• v.41 no.2
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• pp.257-267
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• 2004

In this paper, we introduce, for each approximate distribution $\~{T}$ of [a, b], the approximate Henstock-Stieltjes integral with value in Banach spaces. The Henstock integral is a special case of this where $\~{T}\;=\;\{(\tau,\;[a,\;b])\;:\;{\tau}\;{\in}\;[a,\;b]\}$. This new concept generalizes Henstock integral and abstract Perron-Stieltjes integral. We establish a uniform convergence theorem for approximate Henstock-Stieltjes integral, which is an improvement of the uniform convergence theorem for Perron-Stieltjes integral by Schwabik [3].

• Journal of Korean Society for Quality Management
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• v.33 no.3
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• pp.149-155
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• 2005

Cusum control chart is used for the purpose of controling the process mean. We consider the problem related to cusum chart for controling process variance. Previous researches have considered the same problem. The main difficulty shown in the related researches was to derive the ARL function which characterizes the properties of the chart. Sample variance, differently with sample mean, follows chi-squared type distribution, even when the quality characteristics are assumed to be normally distributed. The ARL function of cusum is described by a type of integral equation. Since the solution of the integral equation for non-normal distribution is not known well, people used simulation method instead of solving the integral equation directly, or approximation method by taking logarithm of the sample variance. Recently a new method to solve the integral equation for Erlang distribution was published. Here we consider the steps to apply the solution to the problem of controling process variance.

• Proceedings of the Korean Society for Quality Management Conference
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• 2006.04a
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• pp.135-141
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• 2006

Cusum control chart is used for the purpose of controling the process mean. We consider the problem related to cusum chart for controling process variance. Previous researches have considered the same problem. The main difficulty shown in the related researches was to derive the ARL function which characterizes the properties of the chart. Sample variance, differently with sample mean, follows chi-squared type distribution, even when the quality characteristics are assumed to be normally distributed. The ARL function of cusum is described by a type of integral equation. Since the solution of the integral equation for non-normal distribution is not known well, people used simulation method instead of solving the integral equation directly, or approximation method by taking logarithm of the sample variance. Recently a new method to solve the integral equation for Erlang distribution was published. Here we consider the steps to apply the solution to the problem of controling process variance.

• Communications for Statistical Applications and Methods
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• v.5 no.2
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• pp.503-515
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• 1998

When a neutral particle beam(NPB) aimed at the object and receive a small number of neutron signals at the detector, it follows approximately Poisson distribution. Under the four assumptions in the presence of errors and uncertainties for the Poisson parameters, an exact probability distribution of neutral particles have been derived. The probability distribution for the neutron signals received by a detector averaged over the three sources of errors is expressed as a four-dimensional integral of certain data. Two of the four integrals can be evaluated analytically and thereby the integral is reduced to a two-dimensional integral. The monotone likelihood ratio(MLR) property of the distribution is proved by using the Cauchy mean value theorem for the univariate distribution and multivariate distribution. Its MLR property can be used to find a criteria for the hypothesis testing problem related to the distribution.

• Communications of the Korean Mathematical Society
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• v.34 no.2
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• pp.383-390
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• 2019

In this paper, we discuss special polynomials involving exponential distribution, which is related to life testing. We derive some identities of special polynomials such as the symmetric property, recurrence formula and so on. In addition, we investigate explicit properties of special polynomials by using their derivative and integral.

• Structural Engineering and Mechanics
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• v.34 no.5
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• pp.635-662
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• 2010

In this study, the effect of superstructure-abutment continuity on the distribution of live load effects among the girders of integral abutment bridges (IABs) is investigated. For this purpose, two and three dimensional finite element models of several single-span, symmetrical integral abutment and simply supported (jointed) bridges (SSBs) are built and analyzed. In the analyses, the effect of various superstructure properties such as span length, number of design lanes, girder size and spacing as well as slab thickness are considered. The results from the analyses of two and three dimensional finite element models are then used to calculate the live load distribution factors (LLDFs) for the girders of IABs and SSBs as a function of the above mentioned parameters. LLDFs for the girders are also calculated using the AASHTO formulae developed for SSBs. Comparison of the analyses results revealed that the superstructure-abutment continuity in IABs produces a better distribution of live load effects among the girders compared to SSBs. The continuity effects become more predominant for short span IABs. Furthermore, AASHTO live load distribution formulae developed for SSBs lead to conservative estimates of live load girder moments and shears for short-span IABs.

• Proceedings of the KOR-KST Conference
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• 1998.09a
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• pp.91-99
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• 1998

Generally, the tons lifted and ton-kilometers are widely used to analyze the road freight transport. However, these two indicators are simply to show the road freight transport statistics rather than to explain the road freight transportation systems. In this study, the variables such as the number of distribution channel levels, the integral distance, tons in transport and the average transport distance are defined and estimated to investigate the road freight transport system of Korea. In order to compare the road transport system of Korea to other countries, the comparative study was conducted including USA, Japan, Holland and Taiwan. The major findings of this study are as follows; i) The number of distribution channel levels and integral distance of Korea and Taiwan have been increased, but the average transport distance decreased from 1971 to 1996 period. ii) On the contrary to Korea case, the number of distribution channel levels and integral distance of US, Japan and Holland have been decreased, but the average transport distance increased. iii) In the time-series model analysis shows that the number of distribution channel levels are statistically positively closely related to the logistics costs and the costs of transportation as a percentage of GDP.

• Journal of the Korean Mathematical Society
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• v.44 no.2
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• pp.455-466
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• 2007

The zero-distribution of the Fourier integral $${\int}^{\infty}_{-{\infty}}\;Q(u)e^{p(u)+^{izu}du$$, where P is a polynomial with leading term $-u^{2m}(m\;{\geq}\;1)$ and Q an arbitrary polynomial, is described. To this end, an asymptotic formula for the integral is established by applying the saddle point method.

• Communications for Statistical Applications and Methods
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• v.22 no.1
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• pp.69-80
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• 2015

In this paper, the probability that a given point is covered by a random convex hull generated by independent and identically-distributed random points in a plane is studied. It is shown that such probability can be expressed in terms of an integral that can be approximated numerically by function-evaluations over the grid-points in a 2-dimensional space. The new integral representation allows such probability be computed efficiently. The computational burdens under the proposed integral representation and those in the existing literature are compared. The proposed method is illustrated through numerical examples where the random points are drawn from (i) uniform distribution over a square and (ii) bivariate normal distribution over the two-dimensional Euclidean space. The applications of the proposed method in statistics are are discussed.