Introduction to Weibull and Exponential Distributions

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Introduction to Weibull and Exponential
Distributions

• Presented by Zack Whitman

Date 4 Mar 09
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Agenda

• Introduce and discuss two statistical
  distributions
• Exponential
• Weibull
• Examples
• Chapter 3, Problem 2 from Practical Reliability
  Engineering by OConnor
• Hopefully provide an understanding about how to
  plot by hand

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References

• NIST/SEMATECH e-Handbook of Statistical Methods,
  http//www.itl.nist.gov/div898/handbook/, 2007
• OConnor, Patrick D. T., Practical Reliability
  Engineering, 4th Edition, John Wiley Sons,
  Ltd., 2002
• Abernethy, R. B., The New Weibull Handbook,
  Robert B. Abernethy, 2005
• ReliaSoft, Life Data Analysis Reference,
  ReliaSoft Publishing, 2005

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Weibull Exponential

• Exponential good in situations of constant
  failure rate appropriate whenever failures
  occur randomly and are not age dependent.
  Blitschke, et al
• Electronics
• acts of God
• Weibull Many desirable properties, common
  modeling distribution for many items (aircraft,
  bearings, human failures,)
• Skewed right
• (B lt 1) infant mortality, (B 1) random
  failures (and equal to the exponential
  distribution), (4 gt B gt 1) early wearout
  failures, (B gt 4) old age, rapid wear out
  Abernethy
• A 3 parameter Weibull uses ? shift to define a
  failure free period

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Exponential Distribution

• PDF CDF

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Weibull Equation

• PDF CDF
• Benards Median Ranking (i - 0.3)/(N 0.4) where
  N number of points and i sorted rank
• i.e. if it was 1 of 3 then its rank would be (1 -
  0.3)/(3 0.4) 0.206
• Note when Slope 1, it is equal to the
  Exponential distribution where ?1/?

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Weibull CDF
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Weibull PDF
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Weibull Plotting Example

• There are six tests for which there are failure
  data. The hours at failure are given for all
  six. Please plot the failures and determine the
  slope and location parameters for these tests.

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Weibull Plotting Example
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Weibull Plotting Example

• A slope of 1.9
• 1.999 exact
• Eta is roughly 1200,
• 1280 exact

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Additional Weibull information

• The mean of a Weibull distribution is
• where the gamma function is defined as
• Reliasoft

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Class Example 1
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Class Example 2
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Class Examples

• Class Example 1, ß5, ?600
• Class Example 2, ß2, ?1,000
• If you wanted a high Median Value, which one
  would you choose?
• If you wanted a high B1 Life, (point at which 1
  fail) which would you choose?
• Next time you get aboard an airplane what B-life
  would you expect/want? B10? B1? B0.1? B0.01?