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

Info icon This preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
Time-Dependent Failure Models 07/04/2015
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
[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
Image of page 2
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 .
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 4
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)
Image of page 5

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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 ?
Image of page 6
Image of page 7
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern