Example: Gumbel Distribution - Revision history

Kate Racaza: Redirected page to Gumbel Distribution Example

2012-08-15T06:42:12Z

Redirected page to Gumbel Distribution Example

← Older revision		Revision as of 06:42, 15 August 2012
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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>

	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\|400px\| ]]~~

	~~Other simulation settings are:~~

	~~[[Image:Gumbel Distribution Example 1 Simulation Setting.png\|thumb\|center\|400px\| ]]~~

	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.

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

	~~::<math>\begin{align}~~
	~~& \hat{\mu }= & 9.3816 \\~~
	~~& \hat{\sigma }= & 1.9717~~
	~~\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.~~

Chris Kahn at 09:31, 26 July 2012

2012-07-26T09:31:18Z

← Older revision		Revision as of 09:31, 26 July 2012
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Chris Kahn at 22:32, 3 May 2012

2012-05-03T22:32:38Z

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	<br>		<br>
	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.
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	[[Image:Gumbel Distribution Example 1 Simulation Setting.png\|thumb\|center\|400px\| ]]		[[Image:Gumbel Distribution Example 1 Simulation Setting.png\|thumb\|center\|400px\| ]]

	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 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.

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

Nicolette Young: Created page with '{| class="FCK__ShowTableBorders" border="0" cellspacing="0" cellpadding="0" align="center"; style="width:100%;" |- | valign="middle" align="left" bgcolor=EEEDF7|[[Image:Weibull-E…'

2012-03-21T23:50:28Z

Created page with '{| class="FCK__ShowTableBorders" border="0" cellspacing="0" cellpadding="0" align="center"; style="width:100%;" |- | valign="middle" align="left" bgcolor=EEEDF7|[[Image:Weibull-E…'

New page

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| valign="middle" align="left" bgcolor=EEEDF7|[[Image:Weibull-Examples-banner.png|400px|center]]
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<br>
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.

[[Image:montecarlo4eva.png|thumb|center|400px| ]]

Other simulation settings are:

[[Image:Gumbel Distribution Example 1 Simulation Setting.png|thumb|center|400px| ]]

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.

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

::<math>\begin{align}
& \hat{\mu }= & 9.3816 \\
& \hat{\sigma }= & 1.9717
\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.