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| | #REDIRECT[[Gumbel_Distribution_Example]] |
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| | valign="middle" align="left" bgcolor=EEEDF7|[[Image:Weibull_Examples_Banner.png|400px|center]]
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| <br>
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| Verify using Monte Carlo simulation that if <math>{{t}_{i}}</math> follows a Weibull distribution with <math>\beta </math> and <math>\eta </math> , then the <math>Ln({{t}_{i}})</math> follows a Gumbel distribution with <math>\mu =\ln (\eta )</math> and <math>\sigma =1/\beta.</math>
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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.
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| [[Image:montecarlo4eva.png|thumb|center|400px| ]]
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| Other simulation settings are:
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| [[Image:Gumbel Distribution Example 1 Simulation Setting.png|thumb|center|400px| ]]
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| After obtaining the random time values <math>{{t}_{i}}</math> , insert a new data sheet into the folio. 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 all 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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| Using maximum likelihood as the analysis method, the estimated parameters are:
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| ::<math>\begin{align}
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| & \hat{\mu }= & 9.3816 \\
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| & \hat{\sigma }= & 1.9717
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| \end{align}</math>
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| 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>
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| Note: This example illustrates a property of the Gumbel distribution; it is not meant to be a formal proof.
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