Gumbel Distribution Example: Difference between revisions

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Let us assume that  <math>{{t}_{i}}</math>  follows a Weibull distribution with  <math>\beta =0.5</math>  and  <math>\eta =10000.</math>  The Monte Carlo simulation tool in Weibull++ can be used to generate a set of random numbers that follow a Weibull distribution with the specified parameters.
Let us assume that  <math>{{t}_{i}}</math>  follows a Weibull distribution with  <math>\beta =0.5</math>  and  <math>\eta =10000.</math>  The Monte Carlo simulation tool in Weibull++ can be used to generate a set of random numbers that follow a Weibull distribution with the specified parameters.


[[Image:montecarlo4eva.png|thumb|center|250px| ]]  
[[Image:montecarlo4eva.png|thumb|center|450px| ]]  


Other simulation settings are:
Other simulation settings are:


[[Image:Gumbel Distribution Example 1 Simulation Setting.png|thumb|center|250px| ]]  
[[Image:Gumbel Distribution Example 1 Simulation Setting.png|thumb|center|450px| ]]  


After obtaining the random time values  <math>{{t}_{i}}</math> , insert a new Data Sheet using the Insert Data Sheet option under the Folio menu. In this sheet enter the  <math>Ln({{t}_{i}})</math>  values using the LN function and referring to the cells in the sheet that contains the  <math>{{t}_{i}}</math>  values. Delete any negative values, if there are any, since Weibull++ expects time values to be positive. Calculate the parameters of the Gumbel distribution that fits the  <math>Ln({{t}_{i}})</math>  values.
After obtaining the random time values  <math>{{t}_{i}}</math> , insert a new Data Sheet using the Insert Data Sheet option under the Folio menu. In this sheet enter the  <math>Ln({{t}_{i}})</math>  values using the LN function and referring to the cells in the sheet that contains the  <math>{{t}_{i}}</math>  values. Delete any negative values, if there are any, since Weibull++ expects time values to be positive. Calculate the parameters of the Gumbel distribution that fits the  <math>Ln({{t}_{i}})</math>  values.
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  & \hat{\sigma }= & 1.9717   
  & \hat{\sigma }= & 1.9717   
\end{align}</math>
\end{align}</math>


Since  <math>\ln (\eta )=</math>  9.2103 ( <math>\simeq 9.3816</math> ) and  <math>1/\beta =2</math>  <math>(\simeq 1.9717),</math>  then this simulation verifies that  <math>Ln({{t}_{i}})</math>  follows a Gumbel distribution with  <math>\mu =\ln (\eta )</math>  and  <math>\delta =1/\beta .</math>  
Since  <math>\ln (\eta )=</math>  9.2103 ( <math>\simeq 9.3816</math> ) and  <math>1/\beta =2</math>  <math>(\simeq 1.9717),</math>  then this simulation verifies that  <math>Ln({{t}_{i}})</math>  follows a Gumbel distribution with  <math>\mu =\ln (\eta )</math>  and  <math>\delta =1/\beta .</math>  


Note: This example illustrates a property of the Gumbel distribution; it is not meant to be a formal proof.
Note: This example illustrates a property of the Gumbel distribution; it is not meant to be a formal proof.

Revision as of 03:45, 6 August 2012

A Gumbel Distribution Example

Verify using Monte Carlo simulation that if [math]\displaystyle{ {{t}_{i}} }[/math] follows a Weibull distribution with [math]\displaystyle{ \beta }[/math] and [math]\displaystyle{ \eta }[/math] , then the [math]\displaystyle{ Ln({{t}_{i}}) }[/math] follows a Gumbel distribution with [math]\displaystyle{ \mu =\ln (\eta ) }[/math] and [math]\displaystyle{ \sigma =1/\beta ). }[/math]

Let us assume that [math]\displaystyle{ {{t}_{i}} }[/math] follows a Weibull distribution with [math]\displaystyle{ \beta =0.5 }[/math] and [math]\displaystyle{ \eta =10000. }[/math] The Monte Carlo simulation tool in Weibull++ can be used to generate a set of random numbers that follow a Weibull distribution with the specified parameters.

Montecarlo4eva.png

Other simulation settings are:

Gumbel Distribution Example 1 Simulation Setting.png

After obtaining the random time values [math]\displaystyle{ {{t}_{i}} }[/math] , insert a new Data Sheet using the Insert Data Sheet option under the Folio menu. In this sheet enter the [math]\displaystyle{ Ln({{t}_{i}}) }[/math] values using the LN function and referring to the cells in the sheet that contains the [math]\displaystyle{ {{t}_{i}} }[/math] values. Delete any negative values, if there are any, since Weibull++ expects time values to be positive. Calculate the parameters of the Gumbel distribution that fits the [math]\displaystyle{ Ln({{t}_{i}}) }[/math] values.

Using maximum likelihood as the analysis method, the estimated parameters are:

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 9.3816 \\ & \hat{\sigma }= & 1.9717 \end{align} }[/math]

Since [math]\displaystyle{ \ln (\eta )= }[/math] 9.2103 ( [math]\displaystyle{ \simeq 9.3816 }[/math] ) and [math]\displaystyle{ 1/\beta =2 }[/math] [math]\displaystyle{ (\simeq 1.9717), }[/math] then this simulation verifies that [math]\displaystyle{ Ln({{t}_{i}}) }[/math] follows a Gumbel distribution with [math]\displaystyle{ \mu =\ln (\eta ) }[/math] and [math]\displaystyle{ \delta =1/\beta . }[/math]

Note: This example illustrates a property of the Gumbel distribution; it is not meant to be a formal proof.