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===Characteristics===
#REDIRECT [[Distributions_Used_in_Accelerated_Testing#The_Weibull_Distribution]]
The characteristics of the 2-parameter Weibull distribution can be exemplified by examining the two parameters, beta,  <math>\beta ,</math>  and eta,  <math>\eta ,</math>  and the effect they have on the  <math>pdf,</math>  reliability and failure rate functions.
 
====Looking at  <math>\beta </math>====
Beta,  <math>\beta ,</math>  is called the shape parameter or slope of the Weibull distribution. Changing the value of  <math>\beta </math>  forces a change in the shape of the  <math>pdf</math>  as shown in Fig. 6. In addition, when the  <math>cdf</math>  is plotted on Weibull probability paper, as shown in Fig. 7, a change in beta is a change in the slope of the distribution on Weibull probability paper.
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'''Effects of  <math>\beta </math>  on the ''pdf'''''
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[[Image:ALTA4.3.gif|thumb|center|400px|Weibull ''pdf'' with <math>0<\beta<1</math>, <math>\Beta=1 </math> and a fixed <math>\eta </math>.]]  
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:• For  <math>0<\beta <1</math> , the failure rate decreases with time and:
::o As  <math>T\to 0,</math>  <math>f(T)\to \infty .</math>
::o As  <math>T\to \infty </math> ,  <math>f(T)\to 0</math> .
::o <math>f(T)</math>  decreases monotonically and is convex as  <math>T</math>  increases.
::o The mode is non-existent.
:• For  <math>\beta =1,</math>  it becomes the exponential distribution, as a special case, <br>
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:or:
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::<math>f(T)=\frac{1}{\eta }{{e}^{-\tfrac{T}{\eta }}};\text{ }\eta >0,T\ge 0</math>
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:where  <math>\tfrac{1}{\eta }=\lambda =</math>  chance, useful life, or failure rate.
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:• For  <math>\beta >1</math> ,  <math>f(T),</math>  the Weibull assumes wear-out type shapes (i.e. the failure rate increases with time) and:
::o <math>f(T)=0</math>  at  <math>T=0</math> .
::o <math>f(T)</math>  increases as  <math>T\to \tilde{T}</math>  (mode) and decreases thereafter.
::o For  <math>\beta =2</math>  it becomes the Rayleigh distribution as a special case. For  <math>\beta <2.6</math>  the Weibull  <math>pdf</math>  is positively skewed (has a right tail), for  <math>2.6<\beta <3.7</math>  its coefficient of skewness approaches zero (no tail); consequently, it may approximate the normal  <math>pdf</math> , and for  <math>\beta >3.7</math>  it is negatively skewed (left tail).
:• The parameter  <math>\beta </math>  is a pure number, i.e. it is dimensionless.
 
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'''Effects of  <math>\beta </math>  on the Reliability Function and the ''cdf'''''
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[[Image:ALTA4.4.gif|thumb|center|400px|Weibull ''cdf'', or Unreliability vs. Time, on Weibull probability plotting paper with <math>0<\Beta<1</math>, <math>\Beta=1 </math> and a fixed <math>\eta </math>.]]
 
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[[Image:ALTA4.5.gif|thumb|center|400px|Weibull 1-''cdf'', or Reliability vs. Time, on linear scales with <math>0<\Beta<1</math>, <math>\Beta=1 </math> and a fixed <math>\eta </math>.]]
 
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:• <math>R(T)</math>  decreases sharply and monotonically for  <math>0<\beta <1</math> , it is convex, and decreases less sharply for the same  <math>\beta </math> .
:• For  <math>\beta =1</math>  and the same  <math>\eta </math> ,  <math>R(T)</math>  decreases monotonically but less sharply than for  <math>0<\beta <1</math> , and is convex.
:• For  <math>\beta >1</math> ,  <math>R(T)</math>  decreases as  <math>T</math>  increases but less sharply than before, and as wear-out sets in, it decreases sharply and goes through an inflection point.
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'''Effects of  <math>\beta </math>  on the Failure Rate Function'''
 
[[Image:ALTA4.6.gif|thumb|center|400px|Weibull Failure Rate vs. Time with  <math>0<\Beta<1 </math>, <math>\Beta=1 </math>, <math>\Beta>1 </math>.]]
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:• The Weibull failure rate for  <math>0<\beta <1</math>  is unbounded at  <math>T=0</math> . The failure rate,  <math>\lambda (T),</math>  decreases thereafter monotonically and is convex, approaching the value of zero as  <math>T\to \infty </math>  or  <math>\lambda (\infty )=0</math> . This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When such behavior is encountered, one or more of the following conclusions can be drawn:
::o Burn-in testing and/or environmental stress screening are not well implemented.
::o There are problems in the production line.
::o Inadequate quality control.
::o Packaging and transit problems.
:• For  <math>\beta =1</math> ,  <math>\lambda (T)</math>  yields a constant value of <math>\tfrac{1}{\eta }</math> , or:
::<math>\lambda (T)=\lambda =\frac{1}{\eta }</math>
This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.
:• For  <math>\beta >1</math> ,  <math>\lambda (T)</math>  increases as  <math>T</math>  increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For  <math>1<\beta <2</math>  the  <math>\lambda (T)</math>  curve is concave, consequently the failure rate increases at a decreasing rate as  <math>T</math>  increases.
:• For  <math>\beta =2</math> , or for the Rayleigh distribution case, the failure rate function is given by:
::<math>\lambda (T)=\frac{2}{\eta }\left( \frac{T}{\eta } \right)</math>
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hence there emerges a straight line relationship between  <math>\lambda (T)</math>  and  <math>T</math> , starting at a value of  <math>\lambda (T)=0</math>  at  <math>T=0</math> , and increasing thereafter with a slope of  <math>\tfrac{2}{{{\eta }^{2}}}</math> . Consequently, the failure rate increases at a constant rate as  <math>T</math>  increases. Furthermore, if  <math>\eta =1</math>  the slope becomes equal to 2, and  <math>\lambda (T)</math>  becomes a straight line which passes through the origin with a slope of 2.
:• When  <math>\beta >2</math>  the  <math>\lambda (T)</math>  curve is convex, with its slope increasing as  <math>T</math>  increases. Consequently, the failure rate increases at an increasing rate as  <math>T</math>  increases, indicating wear-out life.
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====Looking at  <math>\eta </math>====
Eta,  <math>\eta ,</math>  is called the scale parameter of the Weibull distribution. The parameter  <math>\eta </math>  has the same units as  <math>T</math> , such as hours, miles, cycles, actuations, etc.
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[[Image:ALTA4.7.gif|thumb|center|300px| ]]
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:• A change in the scale parameter  <math>\eta </math>  has the same effect on the distribution as a change of the abscissa scale.
::o If  <math>\eta </math>  is increased while  <math>\beta </math>  is kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
::o If  <math>\eta </math>  is decreased while  <math>\beta </math>  is kept the same, the distribution gets pushed in toward the left (i.e. toward its beginning, or 0) and its height increases.
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Latest revision as of 01:56, 16 August 2012

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