Difference between revisions of "Weibull distribution"
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The Weibull distribution is often used to represent failure time in reliability models.  It is similar in shape to the [[Gamma|gamma distribution]], but tends to be less skewed and tail-heavy.  It is a continuous distribution over the positive real numbers.
 
The Weibull distribution is often used to represent failure time in reliability models.  It is similar in shape to the [[Gamma|gamma distribution]], but tends to be less skewed and tail-heavy.  It is a continuous distribution over the positive real numbers.
  
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The Weibull distribution has a cumulative density given by  
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The Weibull distribution has a cumulative density on <math>x\ge 0</math> given by:
  
 
<!--[[Image:Weibull_cdf_eq.jpg]]-->
 
<!--[[Image:Weibull_cdf_eq.jpg]]-->
  
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:<math>F(x) = 1 - exp\left({-\left({x\over{shape}}\right)^{shape}}\right)</math>
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:<math>F(x) = 1 - exp\left({-\left({x\over{scale}}\right)^{shape}}\right)</math>
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 +
and F(x)=0 for x<0.  The density on <math>x\ge 0</math> is given by:
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:<math>f(x) = {{shape}\over{scale}} \left({x\over{scale}}\right)^{shape-1} exp\left(-(x/{scale})^{shape}\right)</math>
  
 
= Library =
 
= Library =

Revision as of 16:54, 26 April 2012


Weibull( shape, scale )

The Weibull distribution is often used to represent failure time in reliability models. It is similar in shape to the gamma distribution, but tends to be less skewed and tail-heavy. It is a continuous distribution over the positive real numbers.

The Weibull distribution has a cumulative density on [math]\displaystyle{ x\ge 0 }[/math] given by:


[math]\displaystyle{ F(x) = 1 - exp\left({-\left({x\over{scale}}\right)^{shape}}\right) }[/math]

and F(x)=0 for x<0. The density on [math]\displaystyle{ x\ge 0 }[/math] is given by:

[math]\displaystyle{ f(x) = {{shape}\over{scale}} \left({x\over{scale}}\right)^{shape-1} exp\left(-(x/{scale})^{shape}\right) }[/math]

Library

Distribution

Example

Weibull(10,4) → Weibull graph.jpg

Parameter Estimation

Suppose you have sampled historic data in Data, indexed by I, and you want to find the parameters for the best-fit Weibull distribution. The parameters can be estimated using a linear regression as follows: 

Index bm := ['b','m'];
Var Fx :=  (Rank(Data,I) - 0.5) / Size(I);
Var Z := Ln(-Ln(1-Fx));
Var fit := Regression(Z,Array(bm,[1,Ln(Data)]),I,bm);
Var shape := fit[bm='m'];
Var b := fit[bm='b'];
Var scale := Exp(-b/shape);
[shape,scale]

See Also

Comments


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