The Lognormal Distribution - Revision history

Lisa Hacker at 23:47, 13 March 2023

2023-03-13T23:47:00Z

← Older revision		Revision as of 23:47, 13 March 2023
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	==Characteristics of the Lognormal Distribution ==		==Characteristics of the Lognormal Distribution ==
	{{:Lognormal Distribution Characteristics}}		{{:Lognormal Distribution Characteristics}}

			==Lognormal Distribution Examples ==
			{{:Lognormal Distribution Examples}}

Lisa Hacker: Replaced content with "{{template:LDABOOK|10|The Lognormal Distribution}} The lognormal distribution is commonly used to model the lives of units whose failure modes are of a fatigue-stress nature. Since this includes most, if not all, mechanical systems, the lognormal distribution can have widespread application. Consequently, the lognormal distribution is a good companion to the Weibull distribution when attempting to model these types of units. As may be surmised by the name, the logno..."

2023-03-08T23:44:49Z

Replaced content with "{{template:LDABOOK|10|The Lognormal Distribution}} The lognormal distribution is commonly used to model the lives of units whose failure modes are of a fatigue-stress nature. Since this includes most, if not all, mechanical systems, the lognormal distribution can have widespread application. Consequently, the lognormal distribution is a good companion to the Weibull distribution when attempting to model these types of units. As may be surmised by the name, the logno..."

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Sharon Honecker: /* Lognormal Probability Density Function */

2017-06-30T17:05:18Z

Lognormal Probability Density Function

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	\end{align}\,\!</math>		\end{align}\,\!</math>

	Taking the derivative yields:		Taking the derivative of the relationship between <math>{t}'\,\!</math> and <math>{t}\,\!</math> yields:

	::<math>d{t}'=\frac{dt}{t}\,\!</math>		::<math>d{t}'=\frac{dt}{t}\,\!</math>

Sharon Honecker: /* RRY Example */ Table cleanup

2016-11-17T02:17:22Z

RRY Example: Table cleanup

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	\text{7} & \text{35} & \text{0}\text{.4651} & \text{3.5553} & \text{-0.0873} & \text{12.6405} & \text{-0.0076}& \text{-0.3102} \\		\text{7} & \text{35} & \text{0}\text{.4651} & \text{3.5553} & \text{-0.0873} & \text{12.6405} & \text{-0.0076}& \text{-0.3102} \\
	\text{8} & \text{40} & \text{0}\text{.5349} & \text{3.6889}& \text{0.0873} & \text{13.6078} & \text{0.0076} & \text{0.3219} \\		\text{8} & \text{40} & \text{0}\text{.5349} & \text{3.6889}& \text{0.0873} & \text{13.6078} & \text{0.0076} & \text{0.3219} \\
	\text{9} & \text{50} & \text{0}\text{.6046} & \text{3.~~912~~} & \text{0.2647} & \text{15.3039} & \text{0.0701} &\text{1.0357} \\		\text{9} & \text{50} & \text{0}\text{.6046} & \text{3.9120} & \text{0.2647} & \text{15.3039} & \text{0.0701} &\text{1.0357} \\
	\text{10} & \text{60} & \text{0}\text{.6742} & \text{4.0943} & \text{0.4512} & \text{16.7637} & \text{0.2036}&\text{1.8474} \\		\text{10} & \text{60} & \text{0}\text{.6742} & \text{4.0943} & \text{0.4512} & \text{16.7637} & \text{0.2036}&\text{1.8474} \\
	\text{11} & \text{70} & \text{0}\text{.7439} & \text{4.2485} & \text{0.6552} & \text{18.0497}& \text{0.4292} & \text{2.7834} \\		\text{11} & \text{70} & \text{0}\text{.7439} & \text{4.2485} & \text{0.6552} & \text{18.0497}& \text{0.4292} & \text{2.7834} \\
	\text{12} & \text{80} & \text{0}\text{.8135} & \text{4.~~382~~} & \text{0.8908} & \text{19.2022} & \text{0.7935} & \text{3.9035} \\		\text{12} & \text{80} & \text{0}\text{.8135} & \text{4.3820} & \text{0.8908} & \text{19.2022} & \text{0.7935} & \text{3.9035} \\
	\text{13} & \text{90} & \text{0}\text{.8830} & \text{4.4998} & \text{1.1901} & \text{20.2483}&\text{1.4163} & \text{5.3552} \\		\text{13} & \text{90} & \text{0}\text{.8830} & \text{4.4998} & \text{1.1901} & \text{20.2483}&\text{1.4163} & \text{5.3552} \\
	\text{14} & \text{100}& \text{1.9517} & \text{4.6052} & \text{1.6619} & \text{21.2076} &\text{2.7619} & \text{7.6533} \\		\text{14} & \text{100}& \text{0}\text{.9517} & \text{4.6052} & \text{1.6619} & \text{21.2076} &\text{2.7619} & \text{7.6533} \\
	\sum_{}^{} & \text{ } & \text{ } & \text{49.222} & \text{0} & \text{183.1531} & \text{11.3646} & \text{10.4473} \\		\sum_{}^{} & \text{ } & \text{ } & \text{49.222} & \text{0} & \text{183.1531} & \text{11.3646} & \text{10.4473} \\

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

2015-12-21T22:55:43Z

Example: LR Bounds on Reliability

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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.10 reliability v sigma.png\|center\|~~350px~~\| ]]		[[Image:WB.10 reliability v sigma.png\|center\|450px\| ]]

	As can be determined from the table, the lowest calculated value for <math>R\,\!</math> is 43.444%, while the highest is 81.508%. These represent the two-sided 75% confidence limits on the reliability at <math>t=65\,\!</math>.		As can be determined from the table, the lowest calculated value for <math>R\,\!</math> is 43.444%, while the highest is 81.508%. These represent the two-sided 75% confidence limits on the reliability at <math>t=65\,\!</math>.

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

2015-12-21T22:55:30Z

Example: LR Bounds on Time

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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.10 time vs sigma.png\|center\|~~350px~~\| ]]		[[Image:WB.10 time vs sigma.png\|center\|450px\| ]]

	As can be determined from the table, the lowest calculated value for <math>t\,\!</math> is 43.634, while the highest is 66.085. These represent the two-sided 75% confidence limits on the time at which reliability is equal to 80%.		As can be determined from the table, the lowest calculated value for <math>t\,\!</math> is 43.634, while the highest is 66.085. These represent the two-sided 75% confidence limits on the time at which reliability is equal to 80%.

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

2015-12-21T22:55:15Z

Example: LR Bounds on Parameters

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	These points are represented graphically in the following contour plot:		These points are represented graphically in the following contour plot:

	[[Image:WB.10 lognormal contour plot.png\|center\|~~350px~~\| ]]		[[Image:WB.10 lognormal 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>{\mu }'\,\!</math> is 4.1145, while the highest is 4.4708. These represent the two-sided 75% confidence limits on this parameter. Since solutions for the equation do not exist for values of <math>{{\sigma'		(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>{\mu }'\,\!</math> is 4.1145, while the highest is 4.4708. These represent the two-sided 75% confidence limits on this parameter. Since solutions for the equation do not exist for values of <math>{{\sigma'

Nicolette Young: /* RRX Example */

2015-12-21T22:54:53Z

RRX Example

← Older revision		Revision as of 22:54, 21 December 2015
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	Using Weibull++ , with the Rank Regression on X option, the results are:		Using Weibull++ , with the Rank Regression on X option, the results are:

	[[Image:Lognormal Distribution Example 3 Data and Result.png\|center\|~~550px~~\| ]]		[[Image:Lognormal Distribution Example 3 Data and Result.png\|center\|650px\| ]]

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

Nicolette Young: /* RRY Example */

2015-12-21T22:54:33Z

RRY Example

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	The above example can be repeated using Weibull++ , using RRY.		The above example can be repeated using Weibull++ , using RRY.

	[[Image:Lognormal Distribution Example 2 Data and Result.png\|center\|~~550px~~\| ]]		[[Image:Lognormal Distribution Example 2 Data and Result.png\|center\|650px\| ]]

	The mean can be obtained from the QCP and both the mean and the standard deviation can be obtained from the Function Wizard.		The mean can be obtained from the QCP and both the mean and the standard deviation can be obtained from the Function Wizard.

Lisa Hacker at 23:47, 23 June 2015

2015-06-23T23:47:21Z

← Older revision		Revision as of 23:47, 23 June 2015
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	==Lognormal Distribution Examples==		==Lognormal Distribution Examples==
	{{:Lognormal Distribution Examples}}		{{:Lognormal Distribution Examples}}

	~~[[Category: Completed Theoretical Review]]~~