Gumbel Distribution Example - Revision history

Lisa Hacker at 21:45, 18 September 2023

2023-09-18T21:45:06Z

← Older revision		Revision as of 21:45, 18 September 2023
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	<noinclude>{{Banner Weibull Examples}}		<noinclude>{{Banner Weibull Examples}}
	''This example appears in the [~~[The_Gumbel~~/~~SEV_Distribution\|~~Life ~~Data Analysis Reference book]~~].''		''This example appears in the [https://help.reliasoft.com/reference/life_data_analysis Life data analysis reference].''

	</noinclude>		</noinclude>

Richard House at 17:57, 26 September 2012

2012-09-26T17:57:26Z

← Older revision		Revision as of 17:57, 26 September 2012
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	</noinclude>		</noinclude>
	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>.		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>.

	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. The following picture shows the Main tab of the Monte Carlo Data Generation utility.		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. The following picture shows the Main tab of the Monte Carlo Data Generation utility.

	[[Image:montecarlo4eva.png\|center\|450px\| ]]		[[Image:montecarlo4eva.png\|center\|450px\| ]]
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	On the Settings tab, set the number of points to '''100''' and click '''Generate'''. This creates a new data sheet in the folio that contains random time values <math>{{t}_{i}}\,\!</math>.		On the Settings tab, set the number of points to '''100''' and click '''Generate'''. This creates a new data sheet in the folio that contains random time values <math>{{t}_{i}}\,\!</math>.

	Insert a new data sheet in the folio and enter the corresponding <math>Ln({{t}_{i}})\,\!</math> values of the time values generated by the Monte Carlo simulation. Delete any negative values, if there are any, because Weibull++ expects the time values to be positive. After obtaining the <math>Ln({{t}_{i}})\,\!</math> values, analyze the data sheet using the Gumbel distribution and the MLE parameter estimation method. The estimated parameters are (your results may vary due to the random numbers generated by simulation):		Insert a new data sheet in the folio and enter the corresponding <math>Ln({{t}_{i}})\,\!</math> values of the time values generated by the Monte Carlo simulation. Delete any negative values, if there are any, because Weibull++ expects the time values to be positive. After obtaining the <math>Ln({{t}_{i}})\,\!</math> values, analyze the data sheet using the Gumbel distribution and the MLE parameter estimation method. The estimated parameters are (your results may vary due to the random numbers generated by simulation):

	::<math>\begin{align}		::<math>\begin{align}
	& \hat{\mu }= & 9.3816 \\		& \hat{\mu }= & 9.3816 \\
	& \hat{\sigma }= & 1.9717		& \hat{\sigma }= & 1.9717
	\end{align}</math>		\end{align}\,\!</math>

	Because <math>\ln (\eta )= 9.2103\,\!</math> ( <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>.		Because <math>\ln (\eta )= 9.2103\,\!</math> ( <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.

Richard House at 17:36, 5 September 2012

2012-09-05T17:36:41Z

← Older revision		Revision as of 17:36, 5 September 2012
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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. The following picture shows the Main tab of the Monte Carlo Data Generation utility.		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. The following picture shows the Main tab of the Monte Carlo Data Generation utility.

	~~<br>~~
	[[Image:montecarlo4eva.png\|center\|450px\| ]]		[[Image:montecarlo4eva.png\|center\|450px\| ]]

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	Insert a new data sheet in the folio and enter the corresponding <math>Ln({{t}_{i}})\,\!</math> values of the time values generated by the Monte Carlo simulation. Delete any negative values, if there are any, because Weibull++ expects the time values to be positive. After obtaining the <math>Ln({{t}_{i}})\,\!</math> values, analyze the data sheet using the Gumbel distribution and the MLE parameter estimation method. The estimated parameters are (your results may vary due to the random numbers generated by simulation):		Insert a new data sheet in the folio and enter the corresponding <math>Ln({{t}_{i}})\,\!</math> values of the time values generated by the Monte Carlo simulation. Delete any negative values, if there are any, because Weibull++ expects the time values to be positive. After obtaining the <math>Ln({{t}_{i}})\,\!</math> values, analyze the data sheet using the Gumbel distribution and the MLE parameter estimation method. The estimated parameters are (your results may vary due to the random numbers generated by simulation):


	::<math>\begin{align}		::<math>\begin{align}
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	& \hat{\sigma }= & 1.9717		& \hat{\sigma }= & 1.9717
	\end{align}</math>		\end{align}</math>


	Because <math>\ln (\eta )= 9.2103\,\!</math> ( <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>.		Because <math>\ln (\eta )= 9.2103\,\!</math> ( <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.

Lisa Hacker at 06:39, 14 August 2012

2012-08-14T06:39:56Z

← Older revision		Revision as of 06:39, 14 August 2012
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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. The following picture shows the Main tab of the Monte Carlo Data Generation utility.		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. The following picture shows the Main tab of the Monte Carlo Data Generation utility.

			<br>
	[[Image:montecarlo4eva.png\|center\|450px\| ]]		[[Image:montecarlo4eva.png\|center\|450px\| ]]

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	On the Settings tab, set the number of points to '''100''' and click '''Generate'''. This creates a new data sheet in the folio that contains random time values <math>{{t}_{i}}\,\!</math>.		On the Settings tab, set the number of points to '''100''' and click '''Generate'''. This creates a new data sheet in the folio that contains random time values <math>{{t}_{i}}\,\!</math>.

	Insert a new data sheet in the folio and enter the corresponding <math>Ln({{t}_{i}})</math> values of the time values generated by the Monte Carlo simulation. Delete any negative values, if there are any, because Weibull++ expects the time values to be positive. After obtaining the <math>Ln({{t}_{i}})\,\!</math> values, analyze the data sheet using the Gumbel distribution and the MLE parameter estimation method. The estimated parameters are (your results may vary due to the random numbers generated by simulation):		Insert a new data sheet in the folio and enter the corresponding <math>Ln({{t}_{i}})\,\!</math> values of the time values generated by the Monte Carlo simulation. Delete any negative values, if there are any, because Weibull++ expects the time values to be positive. After obtaining the <math>Ln({{t}_{i}})\,\!</math> values, analyze the data sheet using the Gumbel distribution and the MLE parameter estimation method. The estimated parameters are (your results may vary due to the random numbers generated by simulation):

Lisa Hacker at 06:39, 14 August 2012

2012-08-14T06:39:26Z

← Older revision		Revision as of 06:39, 14 August 2012
Line 3:		Line 3:

	</noinclude>		</noinclude>
	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>		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>.

	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. The following picture shows the Main tab of the Monte Carlo Data Generation utility.		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. The following picture shows the Main tab of the Monte Carlo Data Generation utility.

	[[Image:montecarlo4eva.png\|center\|450px\| ]]		[[Image:montecarlo4eva.png\|center\|450px\| ]]


	On the Settings tab, set the number of points to '''100''' and click '''Generate'''. This creates a new data sheet in the folio that contains random time values <math>{{t}_{i}}</math>.		On the Settings tab, set the number of points to '''100''' and click '''Generate'''. This creates a new data sheet in the folio that contains random time values <math>{{t}_{i}}\,\!</math>.

	Insert a new data sheet in the folio and enter the corresponding <math>Ln({{t}_{i}})</math> values of the time values generated by the Monte Carlo simulation. Delete any negative values, if there are any, because Weibull++ expects the time values to be positive. After obtaining the <math>Ln({{t}_{i}})</math> values, analyze the data sheet using the Gumbel distribution and the MLE parameter estimation method. The estimated parameters are (your results may vary due to the random numbers generated by simulation):		Insert a new data sheet in the folio and enter the corresponding <math>Ln({{t}_{i}})</math> values of the time values generated by the Monte Carlo simulation. Delete any negative values, if there are any, because Weibull++ expects the time values to be positive. After obtaining the <math>Ln({{t}_{i}})\,\!</math> values, analyze the data sheet using the Gumbel distribution and the MLE parameter estimation method. The estimated parameters are (your results may vary due to the random numbers generated by simulation):


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	Because <math>\ln (\eta )= 9.2103\,\!</math> ( <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>		Because <math>\ln (\eta )= 9.2103\,\!</math> ( <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 formal proof.		Note: This example illustrates a property of the Gumbel distribution; it is not meant to be a formal proof.

Kate Racaza at 00:49, 14 August 2012

2012-08-14T00:49:55Z

← Older revision		Revision as of 00:49, 14 August 2012
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	<noinclude>{{Banner Weibull Examples}}		<noinclude>{{Banner Weibull Examples}}
	''This ~~article~~ appears in the [[The_Gumbel/SEV_Distribution\|Life Data Analysis Reference book]].''		''This example appears in the [[The_Gumbel/SEV_Distribution\|Life Data Analysis Reference book]].''

	</noinclude>		</noinclude>
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	On the Settings tab, set the number of points to '''100''' and click '''Generate'''. This creates a new data sheet in the folio that contains ~~the~~ random time values <math>{{t}_{i}}</math> ~~for the Weibull distribution~~.		On the Settings tab, set the number of points to '''100''' and click '''Generate'''. This creates a new data sheet in the folio that contains random time values <math>{{t}_{i}}</math>.

	Insert a new data sheet in the folio and enter the corresponding <math>Ln({{t}_{i}})</math> values of the time values generated by the Monte Carlo simulation. Delete any negative values, if there are any, because Weibull++ expects the time values to be positive. After obtaining the <math>Ln({{t}_{i}})</math> values, analyze the data sheet using the Gumbel distribution and the MLE parameter estimation method. The estimated parameters are (your results may vary due to the random numbers generated by simulation):		Insert a new data sheet in the folio and enter the corresponding <math>Ln({{t}_{i}})</math> values of the time values generated by the Monte Carlo simulation. Delete any negative values, if there are any, because Weibull++ expects the time values to be positive. After obtaining the <math>Ln({{t}_{i}})</math> values, analyze the data sheet using the Gumbel distribution and the MLE parameter estimation method. The estimated parameters are (your results may vary due to the random numbers generated by simulation):

Kate Racaza: updated example

2012-08-13T07:52:06Z

updated example

← Older revision		Revision as of 07:52, 13 August 2012
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	''~~'A Gumbel Distribution Example'~~''		<noinclude>{{Banner Weibull Examples}}
			''This article appears in the [[The_Gumbel/SEV_Distribution\|Life Data Analysis Reference book]].''

			</noinclude>
	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>		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>

	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. The following picture shows the Main tab of the Monte Carlo Data Generation utility.

	[[Image:montecarlo4eva.png\|center\|450px\| ]]		[[Image:montecarlo4eva.png\|center\|450px\| ]]

	~~Other simulation settings are:~~

	~~[[Image:Gumbel Distribution Example 1 Simulation Setting~~.~~png\|center\|450px\| ]]~~		On the Settings tab, set the number of points to '''100''' and click '''Generate'''. This creates a new data sheet in the folio that contains the random time values <math>{{t}_{i}}</math> for the Weibull distribution.

	~~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.		Insert a new data sheet in the folio and enter the corresponding <math>Ln({{t}_{i}})</math> values of the time values generated by the Monte Carlo simulation. Delete any negative values, if there are any, because Weibull++ expects the time values to be positive. After obtaining the <math>Ln({{t}_{i}})</math> values, analyze the data sheet using the Gumbel distribution and the MLE parameter estimation method. The estimated parameters are (your results may vary due to the random numbers generated by simulation):

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

	::<math>\begin{align}		::<math>\begin{align}
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	\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>

	Note: This example illustrates a property of the Gumbel distribution; it is not meant to be a formal proof.		Because <math>\ln (\eta )= 9.2103\,\!</math> ( <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 formal proof.

Kate Racaza: moved Template:Example: Gumbel distribution example to Gumbel Distribution Example: incorrect namespace

2012-08-13T07:24:15Z

moved Template:Example: Gumbel distribution example to Gumbel Distribution Example: incorrect namespace

← Older revision	Revision as of 07:24, 13 August 2012
(No difference)

Richard House at 05:56, 6 August 2012

2012-08-06T05:56:58Z

← Older revision		Revision as of 05:56, 6 August 2012
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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\|450px\| ]]		[[Image:montecarlo4eva.png\|center\|450px\| ]]

	Other simulation settings are:		Other simulation settings are:

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

Richard House at 03:45, 6 August 2012

2012-08-06T03:45:36Z

← Older revision		Revision as of 03:45, 6 August 2012
Line 5:		Line 5:
	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.
Line 19:		Line 19:
	& \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.