Template:Example: MLE for Exponential Distribution - Revision history

Lisa Hacker: Redirected page to The Exponential Distribution

2012-08-10T08:32:56Z

Redirected page to The Exponential Distribution

← Older revision		Revision as of 08:32, 10 August 2012
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	~~'''MLE for Exponential Distribution'''~~		#REDIRECT [[The Exponential Distribution]]

	~~Using the data of [[Data in Exponential Chapter Example 2\|Example 2]] and assuming a 2-parameter exponential distribution, estimate the parameters using the MLE method.~~

	~~'''Solution'''~~

	~~In this example we have complete data only. The partial derivative of the log-likelihood function, <math>\Lambda ,</math> is given by:~~


	::<math>\frac{\partial \Lambda }{\partial \lambda }=\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,\left[ \frac{1}{\lambda }-\left( {{T}_{i}}-\gamma \right) \right]=\underset{i=1}{\overset{14}{\mathop \sum }}\,\left[ ~~\frac{1}{\lambda }-\left( {{T}_{i}}-\gamma \right) \right]=0</math>~~


	~~Complete descriptions of the partial derivatives can be found in~~ [~~[Appendix:_Log-Likelihood_Equations\|Appendix D]]. Recall that when using the MLE method for the exponential distribution, the value of <math>\gamma </math> is equal to that of the first failure time.~~ The ~~first failure occurred at 5 hours, thus <math>\gamma =5</math> hours<math>.</math> Substituting the values for <math>T</math> and <math>\gamma </math> we get:~~

	~~::<math>\frac{14}{\hat{\lambda }}=560</math>~~

	~~or:~~

	~~::<math>\hat{\lambda }=0.025\text{ failures/hour}</math>~~


	~~Using Weibull++:~~

	~~[[Image:Exponential Distribution Example 4 Data.png\|center\|550px\|]]~~

	~~The probability plot is:~~

	~~[[Image:~~Exponential Distribution ~~Example 4 Plot.png\|center\|550px\|~~]]

Lisa Hacker at 09:30, 6 August 2012

2012-08-06T09:30:47Z

← Older revision		Revision as of 09:30, 6 August 2012
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	'''MLE for Exponential Distribution'''		'''MLE for Exponential Distribution'''

	Using the data of [[Data in Exponential Chapter Example 2\|Example 2]] and assuming a ~~two~~-parameter exponential distribution, estimate the parameters using the MLE method.		Using the data of [[Data in Exponential Chapter Example 2\|Example 2]] and assuming a 2-parameter exponential distribution, estimate the parameters using the MLE method.

	'''Solution'''		'''Solution'''

Lisa Hacker at 09:30, 6 August 2012

2012-08-06T09:30:24Z

← Older revision		Revision as of 09:30, 6 August 2012
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	or:		or:

	::<math>\hat{\lambda }=0.025\text{ failures/hour}.</math>		::<math>\hat{\lambda }=0.025\text{ failures/hour}</math>

Lisa Hacker at 09:30, 6 August 2012

2012-08-06T09:30:09Z

← Older revision		Revision as of 09:30, 6 August 2012
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	Using Weibull++:		Using Weibull++:

	[[Image:Exponential Distribution Example 4 Data.png~~\|thumb~~\|center\|550px\|]]		[[Image:Exponential Distribution Example 4 Data.png\|center\|550px\|]]

	The probability plot is:		The probability plot is:

	[[Image:Exponential Distribution Example 4 Plot.png~~\|thumb~~\|center\|550px\|]]		[[Image:Exponential Distribution Example 4 Plot.png\|center\|550px\|]]

Richard House at 02:40, 6 August 2012

2012-08-06T02:40:26Z

← Older revision		Revision as of 02:40, 6 August 2012
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	Using Weibull++:		Using Weibull++:

	[[Image:Exponential Distribution Example 4 Data.png\|thumb\|center\|~~250px~~\|]]		[[Image:Exponential Distribution Example 4 Data.png\|thumb\|center\|550px\|]]

	The probability plot is:		The probability plot is:

	[[Image:Exponential Distribution Example 4 Plot.png\|thumb\|center\|~~250px~~\|]]		[[Image:Exponential Distribution Example 4 Plot.png\|thumb\|center\|550px\|]]

Lisa Hacker at 09:17, 3 August 2012

2012-08-03T09:17:35Z

← Older revision		Revision as of 09:17, 3 August 2012
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	Complete descriptions of the partial derivatives can be found in [[Appendix: ~~Distribution Log~~-~~Likelihood Equations~~\|Appendix]]. Recall that when using the MLE method for the exponential distribution, the value of <math>\gamma </math> is equal to that of the first failure time. The first failure occurred at 5 hours, thus <math>\gamma =5</math> hours<math>.</math> Substituting the values for <math>T</math> and <math>\gamma </math> we get:		Complete descriptions of the partial derivatives can be found in [[Appendix:_Log-Likelihood_Equations\|Appendix D]]. Recall that when using the MLE method for the exponential distribution, the value of <math>\gamma </math> is equal to that of the first failure time. The first failure occurred at 5 hours, thus <math>\gamma =5</math> hours<math>.</math> Substituting the values for <math>T</math> and <math>\gamma </math> we get:

	::<math>\frac{14}{\hat{\lambda }}=560</math>		::<math>\frac{14}{\hat{\lambda }}=560</math>

Nicolette Young at 17:55, 25 April 2012

2012-04-25T17:55:08Z

← Older revision		Revision as of 17:55, 25 April 2012
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	Using Weibull++:		Using Weibull++:

	[[Image:Exponential Distribution Example 4 Data.png\|thumb\|center\|~~400px~~\|]]		[[Image:Exponential Distribution Example 4 Data.png\|thumb\|center\|250px\|]]

	The probability plot is:		The probability plot is:

	[[Image:Exponential Distribution Example 4 Plot.png\|thumb\|center\|~~400px~~\|]]		[[Image:Exponential Distribution Example 4 Plot.png\|thumb\|center\|250px\|]]

Harry Guo at 21:42, 1 March 2012

2012-03-01T21:42:29Z

← Older revision		Revision as of 21:42, 1 March 2012
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	Using Weibull++:		Using Weibull++:

	[[Image:~~weibullfolio1~~.png\|thumb\|center\|400px\|]]		[[Image:Exponential Distribution Example 4 Data.png\|thumb\|center\|400px\|]]

	The probability plot is:		The probability plot is:

	[[Image:~~weibullfolioplot1~~.png\|thumb\|center\|400px\|]]		[[Image:Exponential Distribution Example 4 Plot.png\|thumb\|center\|400px\|]]

Harry Guo at 18:41, 9 February 2012

2012-02-09T18:41:13Z

← Older revision		Revision as of 18:41, 9 February 2012
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	'''MLE for Exponential Distribution'''		'''MLE for Exponential Distribution'''

	Using the data of [[~~The~~ Exponential ~~Distribution~~\|Example 2]] and assuming a two-parameter exponential distribution, estimate the parameters using the MLE method.		Using the data of [[Data in Exponential Chapter Example 2\|Example 2]] and assuming a two-parameter exponential distribution, estimate the parameters using the MLE method.

	'''Solution'''		'''Solution'''

Harry Guo at 18:24, 9 February 2012

2012-02-09T18:24:00Z

← Older revision		Revision as of 18:24, 9 February 2012
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	'''MLE for Exponential Distribution'''		'''MLE for Exponential Distribution'''

	Using the data of [[~~2 parameter exponential distribution RRY example~~\|Example 2]] and assuming a two-parameter exponential distribution, estimate the parameters using the MLE method.		Using the data of [[The Exponential Distribution\|Example 2]] and assuming a two-parameter exponential distribution, estimate the parameters using the MLE method.

	'''Solution'''		'''Solution'''