The Normal Distribution - Revision history

Lisa Hacker at 18:00, 10 March 2023

2023-03-10T18:00:12Z

Sharon Honecker: /* Probability Plotting */

2017-06-29T19:46:43Z

Probability Plotting

← Older revision		Revision as of 19:46, 29 June 2017
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	::<math>\begin{align}		::<math>\begin{align}
	y=a+bT		y=a+bt
	\end{align}\,\!</math>		\end{align}\,\!</math>

Nicolette Young: /* Example: LR Bounds on Reliability */

2015-12-21T22:41:59Z

Example: LR Bounds on Reliability

← Older revision		Revision as of 22:41, 21 December 2015
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	This data set is represented graphically in the following contour plot:		This data set is represented graphically in the following contour plot:

	[[Image:WB.9 reliability v sigma.png\|center\|~~350px~~\| ]]		[[Image:WB.9 reliability v sigma.png\|center\|500px\| ]]

	As can be determined from the table, the lowest calculated value for <math>R\,\!</math> is 24.776%, while the highest is 68.000%. These represent the 80% two-sided confidence limits on the reliability at <math>t=30\,\!</math>.		As can be determined from the table, the lowest calculated value for <math>R\,\!</math> is 24.776%, while the highest is 68.000%. These represent the 80% two-sided confidence limits on the reliability at <math>t=30\,\!</math>.

Nicolette Young: /* Example: LR Bounds on Time */

2015-12-21T22:41:41Z

Example: LR Bounds on Time

← Older revision		Revision as of 22:41, 21 December 2015
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	This data set is represented graphically in the following contour plot:		This data set is represented graphically in the following contour plot:

	[[Image:WB.9 time v sigma contour.png\|center\|~~550px~~\| ]]		[[Image:WB.9 time v sigma contour.png\|center\|500px\| ]]

	As can be determined from the table, the lowest calculated value for <math>t\,\!</math> is 25.046, while the highest is 39.250. These represent the 80% confidence limits on the time at which reliability is equal to 40%.		As can be determined from the table, the lowest calculated value for <math>t\,\!</math> is 25.046, while the highest is 39.250. These represent the 80% confidence limits on the time at which reliability is equal to 40%.

Nicolette Young: /* Example: LR Bounds on Time */

2015-12-21T22:41:26Z

Example: LR Bounds on Time

← Older revision		Revision as of 22:41, 21 December 2015
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	This data set is represented graphically in the following contour plot:		This data set is represented graphically in the following contour plot:

	[[Image:WB.9 time v sigma contour.png\|center\|~~350px~~\| ]]		[[Image:WB.9 time v sigma contour.png\|center\|550px\| ]]

	As can be determined from the table, the lowest calculated value for <math>t\,\!</math> is 25.046, while the highest is 39.250. These represent the 80% confidence limits on the time at which reliability is equal to 40%.		As can be determined from the table, the lowest calculated value for <math>t\,\!</math> is 25.046, while the highest is 39.250. These represent the 80% confidence limits on the time at which reliability is equal to 40%.

	=====Example: LR Bounds on Reliability=====		=====Example: LR Bounds on Reliability=====
	For the same data set given above in [[The_Normal_Distribution#Example:_LR_Bounds_on_Parameters\|the parameter bounds and time bounds examples]], determine the two-sided 80% confidence bounds on the reliability estimate for <math>t=30\,\!</math>. The ML estimate for the reliability at <math>t=30\,\!</math> is 45.739%.		For the same data set given above in [[The_Normal_Distribution#Example:_LR_Bounds_on_Parameters\|the parameter bounds and time bounds examples]], determine the two-sided 80% confidence bounds on the reliability estimate for <math>t=30\,\!</math>. The ML estimate for the reliability at <math>t=30\,\!</math> is 45.739%.

Nicolette Young: /* Example: LR Bounds on Parameters */

2015-12-21T22:40:41Z

Example: LR Bounds on Parameters

← Older revision		Revision as of 22:40, 21 December 2015
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	This data set is represented graphically in the following contour plot:		This data set is represented graphically in the following contour plot:

	[[Image:WB.9 normal parameter contour plot.png\|center\|~~350px~~\| ]]		[[Image:WB.9 normal parameter contour plot.png\|center\|450px\| ]]

	(Note that this plot is generated with degrees of freedom <math>k=1\,\!</math>, as we are only determining bounds on one parameter. The contour plots generated in Weibull++ are done with degrees of freedom <math>k=2\,\!</math>, for use in comparing both parameters simultaneously.) As can be determined from the table, the lowest calculated value for <math>\sigma \,\!</math> is 7.849, while the highest is 17.909. These represent the two-sided 80% confidence limits on this parameter. Since solutions for the equation do not exist for values of <math>\mu \,\!</math> below 22 or above 35.5, these can be considered the two-sided 80% confidence limits for this parameter. In order to obtain more accurate values for the confidence limits on <math>\mu \,\!</math>, we can perform the same procedure as before, but finding the two values of <math>\mu \,\!</math> that correspond with a given value of <math>\sigma .\,\!</math> Using this method, we find that the two-sided 80% confidence limits on <math>\mu \,\!</math> are 21.807 and 35.793, which are close to the initial estimates of 22 and 35.5.		(Note that this plot is generated with degrees of freedom <math>k=1\,\!</math>, as we are only determining bounds on one parameter. The contour plots generated in Weibull++ are done with degrees of freedom <math>k=2\,\!</math>, for use in comparing both parameters simultaneously.) As can be determined from the table, the lowest calculated value for <math>\sigma \,\!</math> is 7.849, while the highest is 17.909. These represent the two-sided 80% confidence limits on this parameter. Since solutions for the equation do not exist for values of <math>\mu \,\!</math> below 22 or above 35.5, these can be considered the two-sided 80% confidence limits for this parameter. In order to obtain more accurate values for the confidence limits on <math>\mu \,\!</math>, we can perform the same procedure as before, but finding the two values of <math>\mu \,\!</math> that correspond with a given value of <math>\sigma .\,\!</math> Using this method, we find that the two-sided 80% confidence limits on <math>\mu \,\!</math> are 21.807 and 35.793, which are close to the initial estimates of 22 and 35.5.

Nicolette Young: /* MLE Example */

2015-12-21T22:40:23Z

MLE Example

← Older revision		Revision as of 22:40, 21 December 2015
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	The plot of the Weibull++ solution for this example is shown next.		The plot of the Weibull++ solution for this example is shown next.

	[[Image:Normal MLE Plot.png\|center\|~~550px~~\| ]]		[[Image:Normal MLE Plot.png\|center\|650px\| ]]

	==Confidence Bounds==		==Confidence Bounds==

Nicolette Young: /* RRX Example */

2015-12-21T22:40:06Z

RRX Example

← Older revision		Revision as of 22:40, 21 December 2015
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	The plot of the Weibull++ solution for this example is shown next.		The plot of the Weibull++ solution for this example is shown next.

	[[Image:Normal RRX Plot.png\|center\|~~600px~~\| ]]		[[Image:Normal RRX Plot.png\|center\|650px\| ]]

	===Maximum Likelihood Estimation===		===Maximum Likelihood Estimation===

Nicolette Young: /* RRX Example */

2015-12-21T22:39:50Z

RRX Example

← Older revision		Revision as of 22:39, 21 December 2015
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	The plot of the Weibull++ solution for this example is shown next.		The plot of the Weibull++ solution for this example is shown next.

	[[Image:Normal RRX Plot.png\|center\|~~550px~~\| ]]		[[Image:Normal RRX Plot.png\|center\|600px\| ]]

	===Maximum Likelihood Estimation===		===Maximum Likelihood Estimation===

Nicolette Young: /* RRY Example */

2015-12-21T22:39:22Z

RRY Example

← Older revision		Revision as of 22:39, 21 December 2015
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	*Click the Calculate icon located on the Main page.		*Click the Calculate icon located on the Main page.

	[[Image:Normal RRY Setting.png\|center\|~~550px~~\| ]]		[[Image:Normal RRY Setting.png\|center\|650px\| ]]

	The probability plot is shown next.		The probability plot is shown next.

	[[Image:Normal RRY Plot.png\|center\|~~550px~~\| ]]		[[Image:Normal RRY Plot.png\|center\|650px\| ]]

	===Rank Regression on X===		===Rank Regression on X===