b.lect11_relib

b.lect11_relib - Outline Distribution Models Used in...

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Outline Distribution Models Used in Reliability Lecture 11 Chapter 3: Random Variables and Their Distributions M. George Akritas M. George Akritas Lecture 11 Chapter 3: Random Variables and Their Distributio n
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Outline Distribution Models Used in Reliability Distribution Models Used in Reliability Terminology Used in Reliability The Log-normal distribution The Gamma and the χ 2 distributions The Weibull distribution M. George Akritas Lecture 11 Chapter 3: Random Variables and Their Distributio n
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Outline Distribution Models Used in Reliability Terminology Used in Reliability The Log-normal distribution The Gamma and the χ 2 distributions The Weibull distribution I A r.v. that records the life time, or time to failure, of a product is called a life time or failure time, or survival time. I Reliability theory deals with models for failure-time distributions . I If T is the time to failure of some product, the reliability of the product at time t is R T ( t ) = P ( T > t ) = 1 - F T ( t ) = 1 - Z t 0 f T ( x ) dx . M. George Akritas Lecture 11 Chapter 3: Random Variables and Their Distributio n
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Outline Distribution Models Used in Reliability Terminology Used in Reliability The Log-normal distribution The Gamma and the χ 2 distributions The Weibull distribution More Terminology I Viewed as a function, R ( t ) is called reliability function or survival function . I The failure rate hazard function is defined as h ( t ) = f ( t ) R ( t ) . (2.1) I The failure or hazard rate provide information regarding the conditional probability of failure in the small interval ( t , t + Δ t ) given that the product has not failed up to time t . In particular, if T is the time to failure P ( t < T < t + Δ t | T > t ) h T ( t t . (2.2) M. George Akritas Lecture 11 Chapter 3: Random Variables and Their Distributio n
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Outline Distribution Models Used in Reliability Terminology Used in Reliability The Log-normal distribution The Gamma and the χ 2 distributions The Weibull distribution The failure rate determines the distribution I The formula giving the pdf in terms of the failure rate is: f T ( t ) = h T ( t ) exp {- Z t 0 h T ( x ) dx } . (2.3) Example Let the failure time T have the exponential distribution with parameter value λ . Thus, f T ( t ) = λ exp {- λ t } , F T ( t ) = 1 - exp {- λ t } , for t > 0, and f T ( t ) = F T ( t ) = 0, for t < 0. Then show: 1. That the reliability function of T is: R T ( t ) = exp {- λ t } . 2. That the failure rate of T is: h T ( t ) = λ . 3. Relation (2.3) for this particular case. M. George Akritas Lecture 11 Chapter 3: Random Variables and Their Distributio n
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Outline Distribution Models Used in Reliability Terminology Used in Reliability The Log-normal distribution The Gamma and the χ 2 distributions The Weibull distribution Definition A failure-time T is said to have the log-normal distribution if its natural logarithm is normally distributed.
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  • Spring '11
  • hi
  • Probability theory, Exponential distribution, Failure rate, Survival analysis, Weibull distribution, M. George Akritas

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b.lect11_relib - Outline Distribution Models Used in...

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