系統可靠度之Beta機率密度函數估計

令 T 表示系統 (system) 的壽命隨機變數，此系統在時間t的可靠度 (reliability) 定義為，即在時間t 時系統仍可正常使用的機率。在給定時間t時，可靠度亦可視為一具有先驗分配 (prior distribution) 之隨機變數。由於可靠度以不發生失效之機率作為其定義，是介於0與1的機率值，而Beta分配是定義在 [0,1] 上的機率分配，所以考慮將可靠度視為一具有以Beta分配為先驗分配的隨機變數。本研究先針對由兩元件串聯所成之系統，給定兩元件在時間t之可靠度分別為，，時，且互相獨立時，利用平滑迭代演算法估計系統可靠度的機率密度函數。其次，再找出使與的機率密度函數機率密度函數的最小的參數，並以作為系統可靠度之機率密度函數之估計。此估計方法可以得到機率密度函數的參數估計，也可以推廣至所有串聯、並聯、串並聯、與串聯系統可靠度機率密度函數之參數估計。最後，本研究以Alpha System (Duran & Booker, 1988) 為例，給定各元件可靠度之Beta分配，衍生大小為77的樣本，進行系統可靠度Beta分配之參數估計。

關鍵字

可靠度； Beta分配；機率密度函數

並列摘要

The reliability of a system at time t is defined as , where T is the random variable of the lifetime of the system; that is, the probability that the system can be functioned as designed until time t. For a given time t , reliability can treat as a random variable with a prior distribution. Since reliability is defined as the probability of a system to consistently perform and maintain its function without failure, it ranges from 0 to 1. And the domain of a Beta distribution is also between 0 and 1. The reliability of a system is then considered as a random variable with Beta prior. This research focuses on the beta estimation of the system reliability for a series system with two independent components first. Let the distributions of component reliability be , , ; the system reliability be . The smoothed recursive algorithm is applied to estimate the probability density function of . Then the parameters of beta distribution are chosen to minimize the ; and is treated as the estimation of the system reliability. This method can be expanded to the parameters estimation of system reliability of all the series, parallel, series-parallel, and parallel-series system. Finally, the Alpha System (Duran & Booker, 1988) is adopted as an illustration for this approach. Given the Beta distribution of each component, a sample with size 77 is generated to estimate the Beta parameters of the system reliability.

並列關鍵字

reliability ； Beta distribution ； probability density function

參考文獻

郭重明. (1990). 基於貝它方配之系統可靠度. 國立中山大學應用數學研究所論文.

Cheng, M.-Y., Gasser, T., & Hall, P. (1999). Nonparametric Density Estimation under Unimodality and Monotonicity Constraints. Journal of Computational and Graphical Statistics, 8(1), 1-21. doi: 10.1080/10618600.1999.10474798

Duran, B. S., & Booker, J. M. (1988). A Bayes sensitivity analysis when using the beta distribution as a prior IEEE Transactions on Reliability, 37(2), 239-247. doi: 10.1109/24.3748

Jones, M. C., Marron, J. S., & Sheather, S. J. (1996). A Brief Survey of Bandwidth Selection for Density Estimation. Journal of the American Statistical Association, 91(433), 401-407. doi: citeulike-article-id:644908

Parzen, E. (1962). On Estimation of a Probability Density Function and Mode. Ann. Math. Statist, 33(3), 1065-1076.

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系統可靠度之Beta機率密度函數估計

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