Template:2 parameter exponential distribution RRY example - Revision history

Lisa Hacker: Redirected page to The Exponential Distribution

2012-08-09T09:07:22Z

Redirected page to The Exponential Distribution

← Older revision		Revision as of 09:07, 9 August 2012
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	[[~~Category: For Deletion~~]]		#REDIRECT [[The_Exponential_Distribution]]

Lisa Hacker: Replaced content with 'Category: For Deletion'

2012-08-08T06:35:52Z

Replaced content with 'Category: For Deletion'

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	~~'''2-Parameter Exponential RRY Example'''<br>~~		[[Category: For Deletion]]
	~~14 units were being reliability tested and the following life test data were obtained:~~

	~~{\| border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"~~
	\|-
	~~!colspan="2" style="text-align:center"\| Life Test Data~~
	\|-
	~~\|- align="center"~~
	~~!Data point index~~
	~~!Time-to-failure~~
	~~\|- align="center"~~
	~~\|1 \|\|5~~
	~~\|- align="center"~~
	~~\|2 \|\|10~~
	~~\|- align="center"~~
	~~\|3 \|\|15~~
	~~\|- align="center"~~
	~~\|4 \|\|20~~
	~~\|- align="center"~~
	~~\|5 \|\|25~~
	~~\|- align="center"~~
	~~\|6 \|\|30~~
	~~\|- align="center"~~
	~~\|7 \|\|35~~
	~~\|- align="center"~~
	~~\|8 \|\|40~~
	~~\|- align="center"~~
	~~\|9 \|\|50~~
	~~\|- align="center"~~
	~~\|10 \|\|60~~
	~~\|- align="center"~~
	~~\|11 \|\|70~~
	~~\|-align="center"~~
	~~\|12 \|\|80~~
	~~\|- align="center"~~
	~~\|13 \|\|90~~
	~~\|- align="center"~~
	~~\|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'''~~

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

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

	~~<center><math>\begin{matrix}~~
	~~N & t_{i} & F(t_{i}) & y_{i} & t_{i}^{2} & y_{i}^{2} & t_{i} y_{i} \\~~
	~~\text{1} & \text{5} & \text{0}\text{.0483} & \text{-0}\text{.0495} & \text{25} & \text{0}\text{.0025} & \text{-0}\text{.2475} \\~~
	~~\text{2} & \text{10} & \text{0}\text{.1170} & \text{-0}\text{.1244} & \text{100} & \text{0}\text{.0155} & \text{-1}\text{.2443} \\~~
	~~\text{3} & \text{15} & \text{0}\text{.1865} & \text{-0}\text{.2064} & \text{225} & \text{0}\text{.0426} & \text{-3}\text{.0961} \\~~
	~~\text{4} & \text{20} & \text{0}\text{.2561} & \text{-0}\text{.2958} & \text{400} & \text{0}\text{.0875} & \text{-5}\text{.9170} \\~~
	~~\text{5} & \text{25} & \text{0}\text{.3258} & \text{-0}\text{.3942} & \text{625} & \text{0}\text{.1554} & \text{-9}\text{.8557} \\~~
	~~\text{6} & \text{30} & \text{0}\text{.3954} & \text{-0}\text{.5032} & \text{900} & \text{0}\text{.2532} & \text{-15}\text{.0956} \\~~
	~~\text{7} & \text{35} & \text{0}\text{.4651} & \text{-0}\text{.6257} & \text{1225} & \text{0}\text{.3915} & \text{-21}\text{.8986} \\~~
	~~\text{8} & \text{40} & \text{0}\text{.5349} & \text{-0}\text{.7655} & \text{1600} & \text{0}\text{.5860} & \text{-30}\text{.6201} \\~~
	~~\text{9} & \text{50} & \text{0}\text{.6046} & \text{-0}\text{.9279} & \text{2500} & \text{0}\text{.8609} & \text{-46}\text{.3929} \\~~
	~~\text{10} & \text{60} & \text{0}\text{.6742} & \text{-1}\text{.1215} & \text{3600} & \text{1}\text{.2577} & \text{-67}\text{.2883} \\~~
	~~\text{11} & \text{70} & \text{0}\text{.7439} & \text{-1}\text{.3622} & \text{4900} & \text{1}\text{.8456} & \text{-95}\text{.3531} \\~~
	~~\text{12} & \text{80} & \text{0}\text{.8135} & \text{-1}\text{.6793} & \text{6400} & \text{2}\text{.8201} & \text{-134}\text{.3459} \\~~
	~~\text{13} & \text{90} & \text{0}\text{.8830} & \text{-2}\text{.1456} & \text{8100} & \text{4}\text{.6035} & \text{-193}\text{.1023} \\~~
	~~\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} \\~~
	~~\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++.~~
	~~Given the values in the table above, calculate <math>\hat{a}</math> and <math>\hat{b}</math>:~~


	~~::<math>\begin{align}~~
	\hat{b}= & \frac{\underset{i=1}{\overset{14}{\mathop{\sum }}}\,{{t}_{i}}{{y}_{i}}-(\underset{i=1}{\overset{14}{\mathop{\sum }}}\,{{t}_{i}})(\underset{i=1}{\overset{14}{\mathop{\sum }}}\,{{y}_{i}})/14}{\underset{i=1}{\overset{14}{\mathop{\sum }}}\,t_{i}^{2}-{{(\underset{i=1}{\overset{14}{\mathop{\sum }}}\,{{t}_{i}})}^{2}}/14} \\
	\\
	~~\hat{b}= & \frac{-927.4899-(630)(-13.2315)/14}{40,600-{{(630)}^{2}}/14}~~
	~~\end{align}</math>~~

	~~or:~~

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

	~~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>~~

	~~or:~~

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


	~~Therefore:~~

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


	~~and:~~

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

	~~or:~~

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

	~~Then:~~

	~~::<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:~~

	~~::<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.~~

	~~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>~~

	~~[[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.~~

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

Lisa Hacker at 09:27, 6 August 2012

2012-08-06T09:27:58Z

← Older revision		Revision as of 09:27, 6 August 2012
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	~~===Example:~~ 2 Parameter Exponential ~~Distribution~~ RRY~~===~~		'''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:

Lisa Hacker at 09:26, 6 August 2012

2012-08-06T09:26:04Z

← Older revision		Revision as of 09:26, 6 August 2012
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	~~<br><br>'''~~2 Parameter Exponential Distribution RRY~~'''~~		===Example: 2 Parameter Exponential Distribution RRY===
	~~[[2 Paramter Exponential Distribution RRY]]~~		14 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:


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

Lisa Hacker at 08:11, 6 August 2012

2012-08-06T08:11:45Z

← Older revision		Revision as of 08:11, 6 August 2012
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	{\|align="center" border=1 cellspacing=0		{\| border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
	\|-		\|-
	!colspan="2" style="text-align:center"\| ~~Table -~~ Life Test Data		!colspan="2" style="text-align:center"\| Life Test Data
	\|-		\|-
	\|- align="center"		\|- align="center"

Lisa Hacker at 08:09, 6 August 2012

2012-08-06T08:09:07Z

← Older revision		Revision as of 08:09, 6 August 2012
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	{\|align="center" border=1 cellspacing=0		{\|align="center" border=1 cellspacing=0
	\|-		\|-
	\|colspan="2" style="text-align:center"\| Table - Life Test Data		!colspan="2" style="text-align:center"\| Table - Life Test Data
	\|-		\|-
	\|- align="center"		\|- align="center"

Richard House at 06:01, 6 August 2012

2012-08-06T06:01:16Z

← Older revision		Revision as of 06:01, 6 August 2012
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	::<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~~\|thumb~~\|center\|550px\|]]		[[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.
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	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\|550px\|]]		[[Image:Exponential Example 2 Plot.png\|center\|550px\|]]

Richard House at 02:38, 6 August 2012

2012-08-06T02:38:23Z

← Older revision		Revision as of 02:38, 6 August 2012
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	::<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\|thumb\|center\|~~250px~~\|]]		[[Image:Exponential Example 2 Data Folio.png\|thumb\|center\|550px\|]]

	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 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\|~~250px~~\|]]		[[Image:Exponential Example 2 Plot.png\|thumb\|center\|550px\|]]

Richard House at 02:31, 6 August 2012

2012-08-06T02:31:57Z

← Older revision		Revision as of 02:31, 6 August 2012
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	'''Solution'''		'''Solution'''

	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>

Nicolette Young at 17:54, 25 April 2012

2012-04-25T17:54:03Z

← Older revision		Revision as of 17:54, 25 April 2012
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	::<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\|thumb\|center\|~~400px~~\|]]		[[Image:Exponential Example 2 Data Folio.png\|thumb\|center\|250px\|]]

	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.
Line 127:		Line 127:
	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\|250px\|]]