Stat 567:Statistical Reliability
Second Class (September 3, 1997)

Describing Life Distributions. Exponential, Weibull, Gumbel, and Lognormal Distributions

Class Objectives:

• Learn the key concepts used to describe life distributions: survivor or reliability function, hazard rate, cummulative hazard rate
• Learn how the concept of aging is used in reliability modeling
• Learn why the exponential distribution is so important in reliability
• Learn why the Weibull distribution is a natural extension of the exponential distribution
• Learn about the Gumbel and lognormal distributions

Homework Assignments:

• Show that the cumulative hazard function of an increasing hazard (failure) rate (IFR) distribution is convex (i.e., holds water).
  
• Show that if T is a positive random variable with an exponential distribution then:
  
• Show that if T measured in minutes is exponential with hazard rate lambda, the T*=T/60 measured in hours is exponential with hazard rate 60 times lambda.
  
• Show that the Weibull distribution is:
  • IFR if the shape parameter is greater than 1
  • DFR if the shape parameter is less than 1
  • Exponential if the shape parameter is equal to 1
    
• Show that alpha is the 63.2 percentile of the Weibull distribution
  
• Take a look at Military Reliability Standards and Handbooks to learn about MIL-STD-217 and other military reliability standards.
  
• Check the web site: Weibull Data for Parts

Class Outline and Main Points:

• Theoretical arguments in favor of distributions
  • Are not based on the central limit theorem
  • Are based on notions of aging and wear out
  • Are based on extreme-value theory
• Distribution function
  • Cumulative distribution function
  • Survival function
  • Density function
  • Hazard or failure rate function
  • Cumulative hazard function
  • Relationships among these functions
  • Adobe Acrobat PDF showing these relationships
• Hazard (failure) rate and aging
  • Constant hazard (failure) rate implies exponential distribution
  • Increasing hazard (failure) rate and wear out
  • Decreasing hazard (failure) rate and infant mortality
• Exponential distribution
• Weibull and Gumbel distributions
  • Log of Weibull is Gumbel
• Normal and Lognormal distribution

Study Questions:

• Review the intuitive explanations that I gave you of f(t)dt and h(t)dt. What is the difference between these two explanations?
• Why is it sometimes best to estimate the cumulative distribution and cumulative hazard functions rather than the density and hazard rate functions?
• Why is it that the bath-tub model sometimes does not apply when modeling electronic parts? Why is it usually applicable when modeling mechanical parts?
• Why is the Weibull distribution model a natural extension of the exponential distribution model? Why do engineers like it?
• Burn-in is the name given to the practice of running a batch of parts for a period of time before shipping. The purpose is to weed out defective parts. Would the distribution describing the batch of parts be most likely IFR or DFR? Why?
• Why needing to burn-in parts is considered a less-than-ideal engineering practice?
• Why is it that the normal distribution does not play its usual central role with reliability data?
• What are the implications of choosing a Lognormal distribution over a Weibull distribution when modeling life data?
• How is the normal distribution most commonly used in conjunction with reliability data?

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