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==The Weibull Distribution==
#REDIRECT [[Distributions_Used_in_Accelerated_Testing#The_Weibull_Distribution]]
The Weibull distribution is one of the most commonly used distributions in reliability engineering because of the many shapes it attains for various values of  <math>\beta </math>  (slope). It can therefore model a great variety of data and life characteristics [18].
<br>
The 2-parameter Weibull  <math>pdf</math>  is given by:
<br>
::<math>f(T)=\frac{\beta }{\eta }{{\left( \frac{T}{\eta } \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}</math>
 
:where:
 
::<math>f(T)\ge 0,\text{ }T\ge 0,\text{ }\beta >0,\text{ }\eta >0\text{ }</math>
<br>
:and:
:• <math>\eta =</math>  scale parameter.
:• <math>\beta =</math>  shape parameter (or slope).
<br>
===Statistical Properties Summary===
====The Mean or MTTF====
The  mean,  <math>\overline{T},</math>  of the 2-parameter Weibull  <math>pdf</math>  is given by:
<br>
::<math>\overline{T}=\eta \cdot \Gamma \left( \frac{1}{\beta }+1 \right)</math>
<br>
where  <math>\Gamma \left( \tfrac{1}{\beta }+1 \right)</math>  is the gamma function evaluated at the value of  <math>\left( \tfrac{1}{\beta }+1 \right)</math> .
<br>
====The Median====
The median,  <math>\breve{T},</math> 
of the 2-parameter Weibull is given by:
<br>
::<math>\breve{T}=\eta {{\left( \ln 2 \right)}^{\tfrac{1}{\beta }}}</math>
<br>
 
====The Mode====
The mode,  <math>\tilde{T},</math> 
of the 2-parameter Weibull is given by:
<br>
::<math>\tilde{T}=\eta {{\left( 1-\frac{1}{\beta } \right)}^{\tfrac{1}{\beta }}}</math>
<br>
 
====The Standard Deviation====
The standard deviation,  <math>{{\sigma }_{T}},</math> 
of the 2-parameter Weibull is given by:
 
::<math>{{\sigma }_{T}}=\eta \cdot \sqrt{\Gamma \left( \frac{2}{\beta }+1 \right)-\Gamma {{\left( \frac{1}{\beta }+1 \right)}^{2}}}</math>
 
====The  <math>cdf</math>  and the Reliability Function====
The  <math>cdf</math>  of the 2-parameter Weibull distribution is given by:
 
::<math>F(T)=1-{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}</math>
 
The Weibull reliability function is given by:
 
::<math>\begin{align}
  & R(T)= & 1-F(t) \\
& = & {{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}} 
\end{align}</math>
 
====The Conditional Reliability Function====
The Weibull conditional reliability function is given by:
 
::<math>R(T,t)=\frac{R(T+t)}{R(T)}=\frac{{{e}^{-{{\left( \tfrac{T+t}{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}}</math>
 
:or:
 
::<math>R(T,t)={{e}^{-\left[ {{\left( \tfrac{T+t}{\eta } \right)}^{\beta }}-{{\left( \tfrac{T}{\eta } \right)}^{\beta }} \right]}}</math>
 
Eqn. (e137) gives the reliability for a new mission of  <math>t</math>  duration, having already accumulated  <math>T</math>  hours of operation up to the start of this new mission, and the units are checked out to assure that they will start the next mission successfully. (It is called conditional because you can calculate the reliability of a new mission based on the fact that the unit(s) already accumulated  <math>T</math>  hours of operation successfully.)
 
====The Reliable Life====
For the 2-parameter Weibull distribution, the reliable life,  <math>{{T}_{R}}</math> , of a unit for a specified reliability, starting the mission at age zero, is given by:
 
::<math>{{T}_{R}}=\eta \cdot {{\left\{ -\ln \left[ R\left( {{T}_{R}} \right) \right] \right\}}^{\tfrac{1}{\beta }}}</math>
 
This is the life for which the unit will function successfully with a reliability of  <math>R({{T}_{R}})</math> . If  <math>R({{T}_{R}})=0.50</math>  then  <math>{{T}_{R}}=\breve{T}</math>,
the median life, or the life by which half of the units will survive.
<br>
 
====The Failure Rate Function====
The 2-parameter Weibull failure rate function,<math>\lambda (T)</math> ,
is given by:
 
::<math>\lambda \left( T \right)=\frac{f\left( T \right)}{R\left( T \right)}=\frac{\beta }{\eta }{{\left( \frac{T}{\eta } \right)}^{\beta -1}}</math>
 
 
===Characteristics===
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.
<br>
'''Effects of  <math>\beta </math>  on the ''pdf'''''
<br>
[[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>.]]
<br>
<br>
:• 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>
<br>
:or:
<br>
::<math>f(T)=\frac{1}{\eta }{{e}^{-\tfrac{T}{\eta }}};\text{ }\eta >0,T\ge 0</math>
<br>
:where  <math>\tfrac{1}{\eta }=\lambda =</math>  chance, useful life, or failure rate.
<br>
:• 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.
 
<br>
'''Effects of  <math>\beta </math>  on the Reliability Function and the ''cdf'''''
<br>
[[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>.]]
 
<br>
[[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>.]]
 
<br>
:• <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.
<br>
'''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>.]]
<br>
:• 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>
<br>
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.
<br>
 
====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.
<br>
[[Image:ALTA4.7.gif|thumb|center|300px| ]]
<br>
:• 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.
<br>
 
===Parameter Estimation===
The estimates of the parameters of the Weibull distribution can be found graphically on probability plotting paper, or analytically using either least squares or maximum likelihood. (Parameter estimation methods are presented in detail in Appendix B.)
<br>
====Probability Plotting====
One method of calculating the parameters of the Weibull distribution is by using probability plotting. To better illustrate this procedure, consider the following example [18].
 
<br>
====Example 3====
Let's assume six identical units are being reliability tested at the same application and operation stress levels. All of these units fail during the test after operating the following times (in hours),  <math>{{T}_{i}}</math> : 93, 34, 16, 120, 53 and 75.
The steps for determining the parameters of the Weibull  <math>pdf</math>  representing the data, using probability plotting, are as follows:
<br>
:• Rank the times-to-failure in ascending order as shown next.
<br>
<center><math>\begin{matrix}
  \text{Time-to-} & \text{Failure Order Number}  \\
  \text{failure, hrs} & \text{out of a Sample Size of 6}  \\
  \text{16} & \text{1}  \\
  \text{34} & \text{2}  \\
  \text{53} & \text{3}  \\
  \text{75} & \text{4}  \\
  \text{93} & \text{5}  \\
  \text{120} & \text{6}  \\
\end{matrix}</math></center>
<br>
:• Obtain their median rank plotting positions. The times-to-failure, with their corresponding median ranks, are shown next.
<br>
<center><math>\begin{matrix}
  \text{Time-to-} & \text{Median}  \\
  \text{failure, hr} & \text{Rank, }%  \\
  \text{16} & \text{10}\text{.91}  \\
  \text{34} & \text{26}\text{.44}  \\
  \text{53} & \text{42}\text{.14}  \\
  \text{75} & \text{57}\text{.86}  \\
  \text{93} & \text{73}\text{.56}  \\
  \text{120} & \text{89}\text{.10}  \\
\end{matrix}</math></center>
<br>
:• On a Weibull probability paper, plot the times and their corresponding ranks. Fig. 11 displays an example of a Weibull probability paper (the solution is given in Fig. 12).
<br>
[[Image:ALTA4.8.gif|thumb|center|300px|Sample Weibull probability paper.]]
<br>
<br>
:• Draw the best possible straight line through the plotted points (as shown in Fig. 12).
:• Obtain the slope of this line by drawing a line, parallel to the one just obtained, through the slope indicator. This value is the estimate of the shape parameter  <math>\widehat{\beta }</math> . In this case  <math>\widehat{\beta }=1.4</math> .
:• At the  <math>Q(t)=63.2%</math>  ordinate point, draw a straight horizontal line until this line intersects the fitted straight line. Draw a vertical line through this intersection until it crosses the abscissa. The value at the intersection of the abscissa is the estimate of  <math>\widehat{\eta }</math> . For this case  <math>\widehat{\eta }=76</math>  hr. (This is always at 63.2% since  <math>Q(T)=1-{{e}^{-{{(\tfrac{\eta }{\eta })}^{\beta }}}}=1-{{e}^{-1}}=0.632=63.2%).</math>
<br>
[[Image:ALTA4.9.gif|thumb|center|300px|Probability plot for Example 3.]]  
<br>
Now any reliability value for any mission time  <math>t</math>  can be obtained. For example, the reliability for a mission of 15 hr, or any other time, can now be obtained either from the plot or analytically (i.e. using the equations given in Section 5.2.1).
 
To obtain the value from the plot, draw a vertical line from the abscissa, at  <math>t=15</math>  hr, to the fitted line. Draw a horizontal line from this intersection to the ordinate and read  <math>Q(t)</math> , in this case  <math>Q(t=15)=9.8%</math> . Thus,  <math>R(t=15)=1-Q(t)=90.2%</math> . This can also be obtained analytically from the Weibull reliability function since both of the parameters are known.
<br>
::<math>R(t=15)={{e}^{-{{\left( \tfrac{15}{\eta } \right)}^{\beta }}}}={{e}^{-{{\left( \tfrac{15}{76} \right)}^{1.4}}}}=90.2%.</math>
<br>
 
{{mle alta for ed}}

Latest revision as of 01:42, 16 August 2012

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