• Title/Summary/Keyword: cumulative normal distribution

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APPROXIMATION TO THE CUMULATIVE NORMAL DISTRIBUTION USING HYPERBOLIC TANGENT BASED FUNCTIONS

  • Yun, Beong-In
    • Journal of the Korean Mathematical Society
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    • v.46 no.6
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    • pp.1267-1276
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    • 2009
  • This paper presents a method for approximation of the standard normal distribution by using hyperbolic tangent based functions. The presented approximate formula for the cumulative distribution depends on one numerical coefficient only, and its accuracy is admissible. Furthermore, in some particular cases, closed forms of inverse formulas are derived. Numerical results of the present method are compared with those of an existing method.

On the Application of Zp Control Charts for Very Small Fraction of Nonconforming under Non-normal Process (비정규 공정의 극소 불량률 관리를 위한 Zp 관리도 적용 방안 연구)

  • Kim, Jong-Gurl;Choi, Seong-Won;Kim, Hye-Mi;Um, Sang-Joon
    • Journal of Korean Society for Quality Management
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    • v.44 no.1
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    • pp.167-180
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    • 2016
  • Purpose: The problem for the traditional control chart is that it is unable to monitor the very small fraction of nonconforming and the underlying distribution is the normal distribution. $Z_p$ control chart is useful where it controls the vert small fraction on nonconforming. In this study, we will design the $Z_p$ control chart in order to use under non-normal process. Methods: $Z_p$ is calculated not by failure rate based on attribute data but using variable data. Control limit for non-normal $Z_p$ control chart is designed based on ${\alpha}$-risk calculated by cumulative distribution function of Burr distribution. ${\beta}$-risk, which is for performance evaluation, obtains in the Burr distribution's cumulative distribution function and control limit. Results: The control limit for non-normal $Z_p$ control chart is designed based on Burr distribution. The sensitivity can be checked through ARL table and OC curve. Conclusion: Non-normal $Z_p$ control chart is able to control not only the very small fraction of nonconforming, but it is also useful when $Z_p$ distribution is non-normal distribution.

Performance Comparison of Cumulative Incidence Estimators in the Presence of Competing Risks (경쟁위험 하에서의 누적발생함수 추정량 성능 비교)

  • Kim, Dong-Uk;Ahn, Chi-Kyung
    • The Korean Journal of Applied Statistics
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    • v.20 no.2
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    • pp.357-371
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    • 2007
  • For the time-to-failure data with competing risks, cumulative incidence functions (CIFs) are commonly estimated using nonparametric methods. If the cases of events due to the cause of primary interest are infrequent relative to other cause of failure, nonparametric methods may result in rather imprecise estimates for CIF. In such cases, Bryant et al. (2004) suggested to model the cause-specific hazard of primary interest parametrically, while accounting for the other modes of failure using nonparametric estimator. We represented the semiparametric cumulative incidence estimator and extended to the model of Weibull and log-normal distribution. We also conducted simulations to access the performance of the semiparametric cumulative incidence estimators and to investigate the impact of model misspecification in log-normal cause-specific hazard model.

Statistical Characteristics of Response Consistency Parameters in Analytic Hierarchy Process (AHP에서의 응답일관성 모수의 통계적 특성과 활용 방안)

  • 고길곤;이경전
    • Journal of the Korean Operations Research and Management Science Society
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    • v.26 no.4
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    • pp.71-82
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    • 2001
  • Using the computer simulation method, we invest19ate the probability distribution of maximum eigenvalue of pair-wise comparison matrix, which has been used as a parameter for measuring the consistency of responses in analytic hierarchy process (AHP). We show that the shape of the distribution of the maximum eigenvalue is different according to the dimension of the matrix. In addition, we cannot find any evidence that the distribution of the Consistency Index is a Normal distribution, which has been claimed in the Previous literature. Accordingly, we suggest using so called K-index calcu1ated based on the concept of cumulative distribution function lather than based on that of arithmetic mean because the probabilistic distribution cannot be assumed to be a Normal distribution. We interpret the simulation results by comparing them with the suggestion of Saaty[11]. Our results show that using Saaty's value could be too generous when the dimension of the matrix is 3 and strict over 4. Finally, we propose new criteria for measuring the response consistency in AHP.

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Analysis on fatigue life distribution of composite materials (복합재료 피로 수명 분포에 관한 고찰)

  • 황운봉;한경섭
    • Transactions of the Korean Society of Mechanical Engineers
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    • v.12 no.4
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    • pp.790-805
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    • 1988
  • Static strength and fatigue life scattering of glass fiber reinforced epoxy composite materials has been studied. Normal, lognormal, two-parameter and three-parameter Weibull distribution functions are used for strength and one-stress fatigue life distribution. The value of mean fatigue life is analysed using mean fatigue life, mean log fatigue life and expected value of 2 and 3-parameter Weibull distribution functions. Modification on non-statistical cumulative damage models is made in order to interpret the result of two-stress level fatigue life scattering. The comparison results show that 3-parameter Weibull distribution has better predictions in static strength and one-stress level fatigue life distributions. However, no advantage of 3-parameter Weibll distribution is found over 2-parameter Weibull distribution in two-stress level fatigue life predictions. It is found that two-stress level fatigue life prediction by the expanded equal rank assumption is close to the experimental data.

Bivariate ROC Curve and Optimal Classification Function

  • Hong, C.S.;Jeong, J.A.
    • Communications for Statistical Applications and Methods
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    • v.19 no.4
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    • pp.629-638
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    • 2012
  • We propose some methods to obtain optimal thresholds and classification functions by using various cutoff criterion based on the bivariate ROC curve that represents bivariate cumulative distribution functions. The false positive rate and false negative rate are calculated with these classification functions for bivariate normal distributions.

Evaluation of Dowel Bearing Strength of Structural Composite Lumber(SCL) on the Effect of Moisture Content

  • Oh, Sei Chang
    • Journal of the Korean Wood Science and Technology
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    • v.31 no.3
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    • pp.61-69
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    • 2003
  • This study investigated the effect of moisture content and loading direction on dowel bearing strength of two types of SCL. Dowel bearing tests of LVL and PSL were conducted with two different MC level, 7.5% and 19%, and two different oriention, L-direction(loading parallel to grain) and X-direction(loading perpendicular to grain). Most of specimens showed typical load-deformation curves and intersected 5% offset line. Failure modes were classified into two categories; spliting(for L-direction specimens) and peeling(for X-direction specimens). Dowel bearing strength generally decreased with increasing MC. The decreasing rate was more significant in X-directon. ESG also decreased with increasing MC, and the ratio of ESG of 7.5% versus 19% was about 1.47. Dowel bearing strength of LVL and PSL in L-direction was higher than that of X-direction. This results indicated that MC and loading orientation had a significant effect on dowel bearing strength of SCL. The average dowel bearing strength of LVL were higher than that of PSL in each loading direction. Two types of probability distribution model were chosen to quantify strength distribution, normal and 2-parameter weibull distribution. The two models showed good agreement with the data, especially in lower tail of the cumulative distribution. Normal and 2-parameter weibull distribution seemed to proper model of the dowel bearing strength for each MC levels.

The New Criteria of Dissolved Gas Analysis for Oil-Filled Transformers Using a Cumulative Distribution Function

  • Cho, Sung-Min;Kim, Jae-Chul;Kweon, Dong-Jin;Koo, Kyo-Sun
    • Journal of the Korean Institute of Illuminating and Electrical Installation Engineers
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    • v.21 no.9
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    • pp.87-94
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    • 2007
  • This paper presents new criteria for DGA(Dissolved Gases Analysis) using CDF(Cumulative Distribution Function) obtained from the data from the diagnosis of transformers operated in KEPCO over a period of 16 years. Because of differences in operating environments, construction type, oil volume, and other factors, the interpretative criteria of DGA at KEPCO differs from other standards such as IEC-60599, or Rogers and Doernenburg. To suggest the most appropriate criteria, the DGA data from transformers under normal conditions as well as from developing fault transformers were collected. Using these data, this study suggests the limitative gas level of transformers under normal operating conditions and verifies the suitability of the criteria. Because the application of this new criterion to transformers at KEPCO increases the detectable ratio of incipient faults and reduces unnecessary follow-up sampling and analysis, the new criteria yields a more reliable prediction of transformer condition.

An Approximation Approach for A Multi-Product Continuous Review Inventory Problem with Budget Constraint (예산의 계약이 있는 다품종 연속적 재고 관리 문제에서 추정을 통한 해법)

  • Lee, Dong-Ju;Yoo, Jae-Wook;Lee, Moon-Su
    • Journal of Korean Society of Industrial and Systems Engineering
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    • v.31 no.4
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    • pp.134-139
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    • 2008
  • Most approaches for continuous review inventory problem need tables for loss function and cumulative standard normal distribution. Furthermore, it is time-consuming to calculate order quantity (Q) and reorder point (r) iteratively until required values are converged. The purpose of this paper is to develop a direct method to get the solution without any tables. We used approximation approaches for loss function and cumulative standard normal distribution. The proposed method can get the solution directly without any iterative procedure for Q, r and without any tables. The performance of the proposed approach is tested by using numerical examples. The budget constraint of this paper assumes that purchasing costs are paid at the time an order is arrived. This constraint can be easily replaced by capacity constraint or budget constraint in which' purchasing costs are paid at the time an order is placed.