# Solution 9 - Time-Dependent Failure Models[Modification for...

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Time-Dependent Failure Models 07/04/2015

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[Modification: for b , please use the method of MLE instead of method of moments] Solution a. The p.d.f. of the exponential distribution for the failure rate function is ? ?, 𝜆 = 𝜆? −𝜆? . The p.d.f. of 12 observations ? 𝑖 is ? ? 𝑖 , 𝜆 = 𝜆? −𝜆? 𝑖 𝑖 = 1, 2, … 12 . 𝜆 = 12 1159 = 0.010353
b. The likelihood function 𝑙(? 1 , ? 2 , … , ? ? ; 𝜆) is 𝑙 ? 1 , ? 2 , … , ? ? ; 𝜆 = ?(? 1 , 𝜆) ?(? 2 , 𝜆) ? ? ? , 𝜆 = ?(? 𝑖 , 𝜆) ? 𝑖=1 = 𝜆 ? ? −𝜆? 𝑖 = 𝜆 ? ? −𝜆 ? 𝑖 𝑛 𝑖=1 ? 𝑖=1 . The logarithm of the likelihood function is ? ? 1 , ? 2 , … , ? ? ; 𝜆 = 𝑛 log 𝜆 − 𝜆 ? 𝑖 ? 𝑖=1 And 𝜕𝐿 ? 1 ,? 2 ,…,? 𝑛 ;𝜆 𝜕𝜆 = ? 𝜆 ? 𝑖 ? 𝑖=1 = 0 . The “best” estimate of 𝜆 is n/ ? 𝑖 ? 𝑖=1 .

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c. 𝑅 𝑡 = 1 − 𝜆? −𝜆𝜉 ?𝜉 𝑡 0 = ? −𝜆𝑡 𝑅 49 = 0.6021 d. 0 0.5 1 0 50 100 150 200 Reliability Time
Derivation of the reliability function from a known hazard rate function 𝑡 = lim Δ𝑡→0 𝑅 𝑡 −𝑅(𝑡+Δ𝑡) Δ𝑡𝑅 𝑡 = 1 𝑅(𝑡) [− ? ?𝑡 𝑅(𝑡) ] Integrating, 𝜆 𝑡 ? 𝑡 0 𝑡 = −?𝑅(𝑡 ) 𝑅(𝑡 ) 𝑅(𝑡) 1 Where 𝑅 0 = 1 establishes the lower limit in the integral on the right-hand side. Then 𝜆 𝑡 ? 𝑡 0 𝑡 = 𝑙𝑛𝑅 𝑡 Or 𝑅 𝑡 = exp[− 𝜆 𝑡 ? 𝑡 0 𝑡 ] (1)

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The Weibull Distribution One of the most useful probability distributions in reliability is the Weibull. The Weibull failure distribution may be used to model both increasing and decreasing failure rates. It is characterized by a hazard rate function of the form 𝜆 𝑡 = 𝑎𝑡 𝑏 . Which is a power function. The function 𝜆 𝑡 is increasing for a>0, b>0 and is decreasing for a>0, b<0. For mathematical convenience it is better to express 𝜆 𝑡 in the following manner: 𝜆 𝑡 = 𝛽 𝜃 ( 𝑡 𝜃 ) 𝛽−1 𝜃 > 0, 𝛽 > 0, 𝑡 ≥ 0 Using Eq. (1), 𝑹 𝒕 = exp 𝛽 𝜃 𝑡 𝜃 𝛽−1 ?
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