有限區間單峰分配之Beta近似

A Beta approximation of unimodal density in finite interval

胡偉元(Wei-Yuan Hu)

指導教授 ： 李孟峰

國立臺北大學/商學院/統計學系/碩士(2009年)

摘要

可靠度是元件或系統在一特定時間內可以正常作用的機率值。當元件壽命分配未知時，系統可靠度 Rs(t)=P{T≥t} 之估計方法為非參數的估計方法。以抽樣來測試元件，可靠度會隨著的抽樣而有所誤差，為了加以掌握可靠度的變動情形，將可靠度視為一具有 Beta 分配的隨機變數，又稱為可靠度之相信程度 (believed-reliability)。本研究利用數值方法，對於兩元件在時間 t 的可靠度為兩獨立的隨機變數 X~Beta(α1,β1), Y~Beta(α2,β2) 其中 α1, β1, α2, β2 > 0 時，找出與 Z=XY 機率密度函數的 L1-norm 最小的 Beta 機率密度函數，作為 Z=XY 之近似分配，以計算系統之可靠度。本文先討論兩元件串聯的情形，找出可靠度的 Beta 近似，再將其推廣至 n 個元件串聯及並聯的情形。由串聯模式、並聯模式、串並聯模式及並串聯模式所構成之系統，皆可利用上述方法求出總系統可靠度之 Beta 近似。

關鍵字

可靠度 ； 可靠度之相信程度 ； Beta分配 ； Beta近似 ； 數值的

並列摘要

Reliability is the probability that a component or a system can function at designed level during some specified period of time. When the distribution of component lifetime is unknown, nonparametric methods are used to estimate the system reliability, Rs(t)=P{T>t}. Due to the randomization of sampling, the reliability of a component varies from lot to lot. In order to describe further the variation and uncertainty of the reliability, it can be viewed as a random variable, also called believed-reliability, which is often assumed to be distributed as a Beta distribution. The purpose of this research is to present a numerical method to evaluate system believed-reliability by component believed-reliabilities under the circumstance that components distributed as Beta function independently and through the system structure function.

Let the believed-reliabilities of two components at time t be independent random variables X~Beta(α1,β1) and Y~Beta(α2,β2), α1, β1, α2, β2 > 0. This research first discusses the case of a series composed of two components to find the Beta approximation for the believed-reliability, Z=XY by minimizing the L1-norm between the density functions, and then extends it to the case of the series and parallel composed of n components. The Beta approximation of the believed-reliability for the system composed of series, parallel, series-parallel and parallel-series configuration can be derived by the previous methods.

並列關鍵字

reliability ； believed-reliability ； Beta distribution ； Beta approximation ； numerical

參考文獻

[10] 郭重明 (1990)，基於貝它分配之系統可靠度，國立中山大學應用數學研究所論文。

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