Template:Example: 2 Parameter Exponential Distribution RRX - Revision history

Lisa Hacker: Redirected page to The Exponential Distribution

2012-08-10T08:32:13Z

Redirected page to The Exponential Distribution

← Older revision		Revision as of 08:32, 10 August 2012
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	~~'''2 Parameter Exponential Distribution RRX'''~~		#REDIRECT [[The Exponential Distribution]]

	~~Using the data of~~ [[Data in Exponential Chapter Example 2\|Example 2]] and assuming a two-parameter exponential distribution, estimate the parameters and determine the correlation coefficient estimate, <math>\hat{\rho }</math>, using rank regression on X.


	~~''' Solution'''~~

	The ~~table constructed in [[Data in Exponential Chapter Example 2\|Example 2]] applies to this example also. Using the values from this table, we get:~~

	~~::<math>\begin{align}~~
	\hat{b}= & \frac{\underset{i=1}{\overset{14}{\mathop{\sum }}}\,{{t}_{i}}{{y}_{i}}-\tfrac{\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 }}}\,y_{i}^{2}-\tfrac{{{\left( \underset{i=1}{\overset{14}{\mathop{\sum }}}\,{{y}_{i}} \right)}^{2}}}{14}} \\
	\\
	~~\hat{b}= & \frac{-927.4899-(630)(-13.2315)/14}{22.1148-{{(-13.2315)}^{2}}/14}~~
	~~\end{align}</math>~~

	~~or:~~

	~~::<math>\hat{b}=-34.5563</math>~~

	~~and:~~

	~~::<math>\hat{a}=\overline{x}-\hat{b}\overline{y}=\frac{\underset{i=1}{\overset{14}{\mathop{\sum }}}\,{{t}_{i}}}{14}-\hat{b}\frac{\underset{i=1}{\overset{14}{\mathop{\sum }}}\,{{y}_{i}}}{14}</math>~~

	~~or:~~

	~~::<math>\hat{a}=\frac{630}{14}-(-34.5563)\frac{(-13.2315)}{14}=12.3406</math>~~

	~~Therefore:~~

	~~::<math>\hat{\lambda }=-\frac{1}{\hat{b}}=-\frac{1}{(-34.5563)}=0.0289\text{ failures/hour}</math>~~

	~~and:~~

	~~::<math>\hat{\gamma }=\hat{a}=12.3406</math>~~

	~~The correlation coefficient is found to be:~~

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

	Note that the equation for regression on Y is not necessarily the same as that for the regression on X. The only time when the two regression methods yield identical results is when the data lie perfectly on a line. If this were the case, the correlation coefficient would be <math>-1</math>. The negative value of the correlation coefficient is due to the fact that the slope of the exponential probability plot is negative.

	~~This example can be repeated using Weibull++, choosing two-parameter exponential and rank regression on X (RRX) methods for analysis, as shown below.~~
	~~The estimated parameters and the correlation coefficient using Weibull++ were found to be:~~

	~~::<math>\begin{array}{*{35}{l}}~~
	~~\hat{\lambda }= &0.0289 \text{failures/hour} \\~~
	~~\hat{\gamma}= & 12.3395 \text{hours} \\~~
	~~\hat{\rho} = &-0.9679 \\~~
	~~\end{array}</math>~~


	~~[[Image:Exponential Distribution Example 3 Data Folio.png\|center\|550px\|]]~~


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


	~~[[Image:~~Exponential Distribution ~~Example 3 Plot.png\|center\|550px\|~~]]

Richard House at 06:02, 6 August 2012

2012-08-06T06:02:18Z

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	[[Image:Exponential Distribution Example 3 Data Folio.png~~\|thumb~~\|center\|550px\|]]		[[Image:Exponential Distribution Example 3 Data Folio.png\|center\|550px\|]]


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	[[Image:Exponential Distribution Example 3 Plot.png~~\|thumb~~\|center\|550px\|]]		[[Image:Exponential Distribution Example 3 Plot.png\|center\|550px\|]]

Richard House at 02:39, 6 August 2012

2012-08-06T02:39:27Z

← Older revision		Revision as of 02:39, 6 August 2012
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	[[Image:Exponential Distribution Example 3 Data Folio.png\|thumb\|center\|~~250px~~\|]]		[[Image:Exponential Distribution Example 3 Data Folio.png\|thumb\|center\|550px\|]]


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	[[Image:Exponential Distribution Example 3 Plot.png\|thumb\|center\|~~250px~~\|]]		[[Image:Exponential Distribution Example 3 Plot.png\|thumb\|center\|550px\|]]

Richard House at 02:33, 6 August 2012

2012-08-06T02:33:09Z

← Older revision		Revision as of 02:33, 6 August 2012
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	''' Solution'''		''' Solution'''

	The ~~Table~~ constructed in [[Data in Exponential Chapter Example 2\|Example 2]] applies to this example also. Using the values from this table, we get:		The table constructed in [[Data in Exponential Chapter Example 2\|Example 2]] applies to this example also. Using the values from this table, we get:

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

Nicolette Young at 17:54, 25 April 2012

2012-04-25T17:54:38Z

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	[[Image:Exponential Distribution Example 3 Data Folio.png\|thumb\|center\|~~400px~~\|]]		[[Image:Exponential Distribution Example 3 Data Folio.png\|thumb\|center\|250px\|]]


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	[[Image:Exponential Distribution Example 3 Plot.png\|thumb\|center\|~~400px~~\|]]		[[Image:Exponential Distribution Example 3 Plot.png\|thumb\|center\|250px\|]]

Harry Guo at 21:34, 1 March 2012

2012-03-01T21:34:20Z

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	[[Image:Exponential Distribution ~~Exampe~~ 3 Data Folio.png\|thumb\|center\|400px\|]]		[[Image:Exponential Distribution Example 3 Data Folio.png\|thumb\|center\|400px\|]]

Harry Guo at 21:33, 1 March 2012

2012-03-01T21:33:15Z

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	[[Image:Exponential Distribution ~~Exampe~~ 3 Plot.png\|thumb\|center\|400px\|]]		[[Image:Exponential Distribution Example 3 Plot.png\|thumb\|center\|400px\|]]

Harry Guo at 21:31, 1 March 2012

2012-03-01T21:31:57Z

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	[[Image:~~weibullfolio1~~.png\|thumb\|center\|400px\|]]		[[Image:Exponential Distribution Exampe 3 Data Folio.png\|thumb\|center\|400px\|]]


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	[[Image:~~weibullfolioplot1~~.png\|thumb\|center\|400px\|]]		[[Image:Exponential Distribution Exampe 3 Plot.png\|thumb\|center\|400px\|]]

Harry Guo at 21:29, 1 March 2012

2012-03-01T21:29:05Z

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	Using the data of [[Data in Exponential Chapter Example 2\|Example 2]] and assuming a two-parameter exponential distribution, estimate the parameters and determine the correlation coefficient estimate, <math>\hat{\rho }</math>, using rank regression on X.		Using the data of [[Data in Exponential Chapter Example 2\|Example 2]] and assuming a two-parameter exponential distribution, estimate the parameters and determine the correlation coefficient estimate, <math>\hat{\rho }</math>, using rank regression on X.

	~~''' Solution to Example 3'''~~

	Table ~~7.2~~ constructed in Example 2 applies to this example also. Using the values from this table, we get:		''' Solution'''

			The Table constructed in [[Data in Exponential Chapter Example 2\|Example 2]] applies to this example also. Using the values from this table, we get:

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

Harry Guo at 18:36, 9 February 2012

2012-02-09T18:36:59Z

← Older revision		Revision as of 18:36, 9 February 2012
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	'''2 Parameter Exponential Distribution RRX'''		'''2 Parameter Exponential Distribution RRX'''

	Using the data of [[Example 2]] and assuming a two-parameter exponential distribution, estimate the parameters and determine the correlation coefficient estimate, <math>\hat{\rho }</math>, using rank regression on X.		Using the data of [[Data in Exponential Chapter Example 2\|Example 2]] and assuming a two-parameter exponential distribution, estimate the parameters and determine the correlation coefficient estimate, <math>\hat{\rho }</math>, using rank regression on X.

	''' Solution to Example 3'''		''' Solution to Example 3'''