2P Exponential Example - Revision history

Lisa Hacker at 21:45, 18 September 2023

2023-09-18T21:45:36Z

← Older revision		Revision as of 21:45, 18 September 2023
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	<noinclude>{{Banner Weibull Examples}}		<noinclude>{{Banner Weibull Examples}}
	''This example also appears in the [~~[The_Exponential_Distribution#Probability_Plotting\|~~Life ~~Data Analysis Reference book]~~]''.		''This example also appears in the [https://help.reliasoft.com/reference/life_data_analysis Life data analysis reference]''.
	</noinclude>		</noinclude>
	'''2-Parameter Exponential RRY Example'''<br>		'''2-Parameter Exponential RRY Example'''<br>

Nicolette Young at 17:12, 16 December 2015

2015-12-16T17:12:43Z

← Older revision		Revision as of 17:12, 16 December 2015
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	This example can be repeated using Weibull++, choosing 2-parameter exponential and rank regression on Y (RRY), as shown next.		This example can be repeated using Weibull++, choosing 2-parameter exponential and rank regression on Y (RRY), as shown next.

	[[Image:Exponential Example 2 Data Folio.png\|center\|~~550px~~\|]]		[[Image:Exponential Example 2 Data Folio.png\|center\|750px\|]]

	The estimated parameters and the correlation coefficient using Weibull++ were found to be:		The estimated parameters and the correlation coefficient using Weibull++ were found to be:

Richard House at 18:10, 25 September 2012

2012-09-25T18:10:07Z

← Older revision		Revision as of 18:10, 25 September 2012
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	Construct the following table, as shown next.		Construct the following table, as shown next.

	<center><math>\overset{{}}{\mathop{\text{Table}\text{- Least Squares Analysis}}}\,</math></center>		<center><math>\overset{{}}{\mathop{\text{Table}\text{- Least Squares Analysis}}}\,\,\!</math></center>

	<center><math>\begin{matrix}		<center><math>\begin{matrix}
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	\text{14} & \text{100} & \text{0}\text{.9517} & \text{-3}\text{.0303} & \text{10000} & \text{9}\text{.1829} & \text{-303}\text{.0324} \\		\text{14} & \text{100} & \text{0}\text{.9517} & \text{-3}\text{.0303} & \text{10000} & \text{9}\text{.1829} & \text{-303}\text{.0324} \\
	\sum_{}^{} & \text{630} & {} & \text{-13}\text{.2315} & \text{40600} & \text{22}\text{.1148} & \text{-927}\text{.4899} \\		\sum_{}^{} & \text{630} & {} & \text{-13}\text{.2315} & \text{40600} & \text{22}\text{.1148} & \text{-927}\text{.4899} \\
	\end{matrix}</math></center>		\end{matrix}\,\!</math></center>


	The median rank values ( <math>F({{t}_{i}})</math> ) can be found in rank tables or they can be estimated using the '''Quick Statistical Reference''' in Weibull++.		The median rank values ( <math>F({{t}_{i}})\,\!</math> ) can be found in rank tables or they can be estimated using the '''Quick Statistical Reference''' in Weibull++.
	Given the values in the table above, calculate <math>\hat{a}</math> and <math>\hat{b}</math>:		Given the values in the table above, calculate <math>\hat{a}\,\!</math> and <math>\hat{b}\,\!</math>:


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	\\		\\
	\hat{b}= & \frac{-927.4899-(630)(-13.2315)/14}{40,600-{{(630)}^{2}}/14}		\hat{b}= & \frac{-927.4899-(630)(-13.2315)/14}{40,600-{{(630)}^{2}}/14}
	\end{align}</math>		\end{align}\,\!</math>

	or:		or:

	::<math>\hat{b}=-0.02711</math>		::<math>\hat{b}=-0.02711\,\!</math>

	and:		and:

	::<math>\hat{a}=\overline{y}-\hat{b}\overline{t}=\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{y}_{i}}}{N}-\hat{b}\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{t}_{i}}}{N}</math>		::<math>\hat{a}=\overline{y}-\hat{b}\overline{t}=\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{y}_{i}}}{N}-\hat{b}\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{t}_{i}}}{N}\,\!</math>

	or:		or:

	::<math>\hat{a}=\frac{-13.2315}{14}-(-0.02711)\frac{630}{14}=0.2748</math>		::<math>\hat{a}=\frac{-13.2315}{14}-(-0.02711)\frac{630}{14}=0.2748\,\!</math>


	Therefore:		Therefore:

	::<math>\hat{\lambda }=-\hat{b}=-(-0.02711)=0.02711\text{ failures/hour}</math>		::<math>\hat{\lambda }=-\hat{b}=-(-0.02711)=0.02711\text{ failures/hour}\,\!</math>


	and:		and:

	::<math>\hat{\gamma }=\frac{\hat{a}}{\hat{\lambda }}=\frac{0.2748}{0.02711}</math>		::<math>\hat{\gamma }=\frac{\hat{a}}{\hat{\lambda }}=\frac{0.2748}{0.02711}\,\!</math>

	or:		or:

	::<math>\hat{\gamma }=10.1365\text{ hours}</math>		::<math>\hat{\gamma }=10.1365\text{ hours}\,\!</math>

	Then:		Then:

	::<math>f(t)=(0.02711)\cdot {{e}^{-0.02711(T-10.136)}}</math>		::<math>f(t)=(0.02711)\cdot {{e}^{-0.02711(T-10.136)}}\,\!</math>


	The correlation coefficient can be estimated using equation for calculating the correlation coefficient:		The correlation coefficient can be estimated using equation for calculating the correlation coefficient:

	::<math>\hat{\rho }=-0.9679</math>		::<math>\hat{\rho }=-0.9679\,\!</math>


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	The estimated parameters and the correlation coefficient using Weibull++ were found to be:		The estimated parameters and the correlation coefficient using Weibull++ were found to be:

	::<math>\hat{\lambda }=0.0271\text{ fr/hr },\hat{\gamma }=10.1348\text{ hr },\hat{\rho }=-0.9679</math>		::<math>\hat{\lambda }=0.0271\text{ fr/hr },\hat{\gamma }=10.1348\text{ hr },\hat{\rho }=-0.9679\,\!</math>

	Please note that the user must deselect the '''Reset if location parameter > T1 on Exponential''' option on the Calculations page of the Application Setup window.		Please note that the user must deselect the '''Reset if location parameter > T1 on Exponential''' option on the Calculations page of the Application Setup window.

Lisa Hacker at 00:12, 16 August 2012

2012-08-16T00:12:04Z

← Older revision		Revision as of 00:12, 16 August 2012
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	</noinclude>		</noinclude>
	'''2-Parameter Exponential RRY Example'''<br>		'''2-Parameter Exponential RRY Example'''<br>
	14 units were being reliability tested and the following life test data were obtained:		14 units were being reliability tested and the following life test data were obtained. Assuming that the data follow a 2-parameter exponential distribution, estimate the parameters and determine the correlation coefficient, <math>\rho \,\!</math>, using rank regression on Y (RRY).

	{\| border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"		{\| border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
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	\|14 \|\|100		\|14 \|\|100
	\|}		\|}

	~~Assuming that the data follow a two-parameter exponential distribution, estimate the parameters and determine the correlation coefficient, <math>\rho </math>, using rank regression on Y.~~

	'''Solution'''		'''Solution'''
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	This example can be repeated using Weibull++, choosing ~~two~~-parameter exponential and rank regression on Y (RRY), as shown next.		This example can be repeated using Weibull++, choosing 2-parameter exponential and rank regression on Y (RRY), as shown next.

	[[Image:Exponential Example 2 Data Folio.png\|center\|550px\|]]		[[Image:Exponential Example 2 Data Folio.png\|center\|550px\|]]

Kate Racaza at 05:06, 15 August 2012

2012-08-15T05:06:54Z

← Older revision		Revision as of 05:06, 15 August 2012
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	<noinclude>{{Banner Weibull Examples}}~~[[Category:NTI]]~~		<noinclude>{{Banner Weibull Examples}}
	''This example also appears in the [[The_Exponential_Distribution#Probability_Plotting\|Life Data Analysis Reference book]]''.		''This example also appears in the [[The_Exponential_Distribution#Probability_Plotting\|Life Data Analysis Reference book]]''.
	</noinclude>		</noinclude>

Melinda Caroline at 05:35, 14 August 2012

2012-08-14T05:35:15Z

← Older revision		Revision as of 05:35, 14 August 2012
Line 116:		Line 116:


	This example can be repeated using Weibull++, choosing two-parameter exponential and rank regression on Y (RRY), as shown ~~in the figure on the following page~~.		This example can be repeated using Weibull++, choosing two-parameter exponential and rank regression on Y (RRY), as shown next.

			[[Image:Exponential Example 2 Data Folio.png\|center\|550px\|]]

	The estimated parameters and the correlation coefficient using Weibull++ were found to be:		The estimated parameters and the correlation coefficient using Weibull++ were found to be:

	::<math>\hat{\lambda }=0.0271\text{ fr/hr },\hat{\gamma }=10.1348\text{ hr },\hat{\rho }=-0.9679</math>		::<math>\hat{\lambda }=0.0271\text{ fr/hr },\hat{\gamma }=10.1348\text{ hr },\hat{\rho }=-0.9679</math>

	~~[[Image:Exponential Example 2 Data Folio.png\|center\|550px\|]]~~

	Please note that the user must deselect the '''Reset if location parameter > T1 on Exponential''' option on the Calculations page of the Application Setup window.		Please note that the user must deselect the '''Reset if location parameter > T1 on Exponential''' option on the Calculations page of the Application Setup window.

Lisa Hacker at 06:34, 8 August 2012

2012-08-08T06:34:38Z

← Older revision		Revision as of 06:34, 8 August 2012
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	<noinclude>{{Banner Weibull Examples}}[[Category:NTI]]</noinclude>		<noinclude>{{Banner Weibull Examples}}[[Category:NTI]]
	'''2 Parameter Exponential ~~Distribution~~ RRY'''		''This example also appears in the [[The_Exponential_Distribution#Probability_Plotting\|Life Data Analysis Reference book]]''.
			</noinclude>
			'''2-Parameter Exponential RRY Example'''<br>
			14 units were being reliability tested and the following life test data were obtained:

	~~Fourteen units were reliability tested and the following life test data were obtained:~~		{\| border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"


	{\|align="center" border=1 cellspacing=0
	\|-		\|-
	\|colspan="2" style="text-align:center"\| ~~Table -~~ Life Test Data		!colspan="2" style="text-align:center"\| Life Test Data
	\|-		\|-
	\|- align="center"		\|- align="center"
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	\|14 \|\|100		\|14 \|\|100
	\|}		\|}


	Assuming that the data follow a two-parameter exponential distribution, estimate the parameters and determine the correlation coefficient, <math>\rho </math>, using rank regression on Y.		Assuming that the data follow a two-parameter exponential distribution, estimate the parameters and determine the correlation coefficient, <math>\rho </math>, using rank regression on Y.


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

	Construct a table, as shown next.		Construct the following table, as shown next.

	<center><math>\overset{{}}{\mathop{\text{Table}\text{- Least Squares Analysis}}}\,</math></center>		<center><math>\overset{{}}{\mathop{\text{Table}\text{- Least Squares Analysis}}}\,</math></center>
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	The median rank values ( <math>F({{t}_{i}})</math> ) can be found in rank tables or they can be estimated using the '''Quick Statistical Reference ~~(QSR)~~''' ~~tool~~ in Weibull++.		The median rank values ( <math>F({{t}_{i}})</math> ) can be found in rank tables or they can be estimated using the '''Quick Statistical Reference''' in Weibull++.
	Given the values in the table above, calculate <math>\hat{a}</math> and <math>\hat{b}</math>:		Given the values in the table above, calculate <math>\hat{a}</math> and <math>\hat{b}</math>:

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	The correlation coefficient can be estimated using ~~the~~ equation for calculating the correlation coefficient:		The correlation coefficient can be estimated using equation for calculating the correlation coefficient:

	::<math>\hat{\rho }=-0.9679</math>		::<math>\hat{\rho }=-0.9679</math>


	This example can be repeated using Weibull++ by choosing ~~the~~ two-parameter exponential ~~distribution~~ and ~~the~~ rank regression on Y (RRY) ~~parameter estimation method. The estimated parameters and the correlation coefficient are <math>\hat{\lambda }=0.0271\text{ fr/hr },\hat{\gamma }=10.1348\text{ hr },\hat{\rho }=-0.9679</math>~~, as shown ~~next~~.		This example can be repeated using Weibull++, choosing two-parameter exponential and rank regression on Y (RRY), as shown in the figure on the following page.

	~~[[Image~~:~~Exponential Example 2 Data Folio.png\|thumb\|center\|400px\|]]~~		The estimated parameters and the correlation coefficient using Weibull++ were found to be:

			::<math>\hat{\lambda }=0.0271\text{ fr/hr },\hat{\gamma }=10.1348\text{ hr },\hat{\rho }=-0.9679</math>

	~~Please note that the user must deselect the '''Reset if Loc. Param > T1 on Exp RR''' option on the user setup page~~.		[[Image:Exponential Example 2 Data Folio.png\|center\|550px\|]]

			Please note that the user must deselect the '''Reset if location parameter > T1 on Exponential''' option on the Calculations page of the Application Setup window.

	The probability plot can be obtained simply by clicking the '''Plot''' icon.		The probability plot can be obtained simply by clicking the '''Plot''' icon.

	[[Image:Exponential Example 2 Plot.png~~\|thumb~~\|center\|~~400px~~\|]]		[[Image:Exponential Example 2 Plot.png\|center\|550px\|]]

Kate Racaza: Started review. Will need theory input

2012-07-20T10:57:13Z

Started review. Will need theory input

← Older revision		Revision as of 10:57, 20 July 2012
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	{{Banner Weibull Examples}}		<noinclude>{{Banner Weibull Examples}}[[Category:NTI]]</noinclude>

	'''2 Parameter Exponential Distribution RRY'''		'''2 Parameter Exponential Distribution RRY'''

	Fourteen units were ~~being~~ reliability tested and the following life test data were obtained:		Fourteen units were reliability tested and the following life test data were obtained:


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	\|14 \|\|100		\|14 \|\|100
	\|}		\|}


	Assuming that the data follow a two-parameter exponential distribution, estimate the parameters and determine the correlation coefficient, <math>\rho </math>, using rank regression on Y.		Assuming that the data follow a two-parameter exponential distribution, estimate the parameters and determine the correlation coefficient, <math>\rho </math>, using rank regression on Y.


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

	Construct ~~the following Table~~, as shown next.		Construct a table, as shown next.

	<center><math>\overset{{}}{\mathop{\text{Table}\text{- Least Squares Analysis}}}\,</math></center>		<center><math>\overset{{}}{\mathop{\text{Table}\text{- Least Squares Analysis}}}\,</math></center>
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	The median rank values ( <math>F({{t}_{i}})</math> ) can be found in rank tables or they can be estimated using the '''Quick Statistical Reference''' in Weibull++.		The median rank values ( <math>F({{t}_{i}})</math> ) can be found in rank tables or they can be estimated using the '''Quick Statistical Reference (QSR)''' tool in Weibull++.
	Given the values in the table above, calculate <math>\hat{a}</math> and <math>\hat{b}</math>:		Given the values in the table above, calculate <math>\hat{a}</math> and <math>\hat{b}</math>:

Line 112:		Line 113:


	The correlation coefficient can be estimated using equation for calculating the correlation coefficient:		The correlation coefficient can be estimated using the equation for calculating the correlation coefficient:

	::<math>\hat{\rho }=-0.9679</math>		::<math>\hat{\rho }=-0.9679</math>


	This example can be repeated using Weibull++, choosing two-parameter exponential and rank regression on Y (RRY), as shown ~~in the figure on the following page~~.		This example can be repeated using Weibull++ by choosing the two-parameter exponential distribution and the rank regression on Y (RRY) parameter estimation method. The estimated parameters and the correlation coefficient are <math>\hat{\lambda }=0.0271\text{ fr/hr },\hat{\gamma }=10.1348\text{ hr },\hat{\rho }=-0.9679</math>, as shown next.

	~~The estimated parameters and the correlation coefficient using Weibull++ were found to be~~:		[[Image:Exponential Example 2 Data Folio.png\|thumb\|center\|400px\|]]

	~~::<math>\hat{\lambda }=0.0271\text{ fr/hr },\hat{\gamma }=10.1348\text{ hr },\hat{\rho }=-0.9679</math>~~

	~~[[Image:Exponential Example 2 Data Folio~~.~~png\|thumb\|center\|400px\|]]~~		Please note that the user must deselect the '''Reset if Loc. Param > T1 on Exp RR''' option on the user setup page.

	~~Please note that the user must deselect the '''Reset if Loc. Param > T1 on Exp RR''' option on the user setup page.~~

	The probability plot can be obtained simply by clicking the '''Plot''' icon.		The probability plot can be obtained simply by clicking the '''Plot''' icon.

	[[Image:Exponential Example 2 Plot.png\|thumb\|center\|400px\|]]		[[Image:Exponential Example 2 Plot.png\|thumb\|center\|400px\|]]

Lisa Hacker at 12:05, 17 July 2012

2012-07-17T12:05:58Z

← Older revision		Revision as of 12:05, 17 July 2012
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	[[Image:Exponential Example 2 Plot.png\|thumb\|center\|400px\|]]		[[Image:Exponential Example 2 Plot.png\|thumb\|center\|400px\|]]

	~~[[Category:Weibull++ Examples]]~~

Lisa Hacker at 11:54, 17 July 2012

2012-07-17T11:54:47Z

← Older revision		Revision as of 11:54, 17 July 2012
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	{{Banner Weibull Examples}}		{{Banner Weibull Examples}}

	~~<br><br>~~'''2 Parameter Exponential Distribution RRY'''		'''2 Parameter Exponential Distribution RRY'''

	Fourteen units were being reliability tested and the following life test data were obtained:		Fourteen units were being reliability tested and the following life test data were obtained: