The Weibull Distribution - Revision history

Lisa Hacker at 15:36, 22 May 2023

2023-05-22T15:36:01Z

← Older revision		Revision as of 15:36, 22 May 2023
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	{{template:LDABOOK\|8\|The Weibull Distribution}}		{{template:LDABOOK\|8\|The Weibull Distribution}}
	The Weibull distribution is one of the most widely used lifetime distributions in reliability engineering. It is a versatile distribution that can take on the characteristics of other types of distributions, based on the value of the shape parameter, <math> {\beta} \,\!</math>. This chapter provides a brief background on the Weibull distribution, presents and derives most of the applicable equations and presents examples calculated both manually and by using ReliaSoft's [https://~~koi-3QN72QORVC~~.~~marketingautomation~~.~~services~~/~~net~~/~~m?md=Rw01CJDOxn%2FabhkPlZsy6DwBQ%2BaCXsGR~~ Weibull++ software].		The Weibull distribution is one of the most widely used lifetime distributions in reliability engineering. It is a versatile distribution that can take on the characteristics of other types of distributions, based on the value of the shape parameter, <math> {\beta} \,\!</math>. This chapter provides a brief background on the Weibull distribution, presents and derives most of the applicable equations and presents examples calculated both manually and by using ReliaSoft's [https://www.hbkworld.com/en/products/software/analysis-simulation/reliability/weibull-life-data-analysis-software Weibull++ software].

	== Weibull Probability Density Function ==		== Weibull Probability Density Function ==

Lisa Hacker at 23:44, 13 March 2023

2023-03-13T23:44:11Z

← Older revision		Revision as of 23:44, 13 March 2023
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	== Characteristics of the Weibull Distribution ==		== Characteristics of the Weibull Distribution ==
	{{:Weibull Distribution Characteristics}}		{{:Weibull Distribution Characteristics}}

			== Weibull Distribution Examples ==
			{{:Weibull Distribution Examples}}

Lisa Hacker at 00:29, 9 March 2023

2023-03-09T00:29:54Z

← Older revision		Revision as of 00:29, 9 March 2023
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	== Characteristics of the Weibull Distribution ==		== Characteristics of the Weibull Distribution ==
	{{:Weibull Distribution Characteristics}}		{{:Weibull Distribution Characteristics}}

	~~== Weibull Distribution Examples ==~~
	~~{{:Weibull Distribution Examples}}~~

Lisa Hacker at 00:27, 9 March 2023

2023-03-09T00:27:47Z

← Older revision		Revision as of 00:27, 9 March 2023
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	== Characteristics of the Weibull Distribution ==		== Characteristics of the Weibull Distribution ==
	{{:Weibull Distribution Characteristics}}		{{:Weibull Distribution Characteristics}}

			== Weibull Distribution Examples ==
			{{:Weibull Distribution Examples}}

Lisa Hacker: Replaced content with "{{template:LDABOOK|8|The Weibull Distribution}} The Weibull distribution is one of the most widely used lifetime distributions in reliability engineering. It is a versatile distribution that can take on the characteristics of other types of distributions, based on the value of the shape parameter, ${\beta} \,\!$ . This chapter provides a brief background on the Weibull distribution, presents and derives most of the applicable equations and presents example..."

2023-03-09T00:03:42Z

Replaced content with "{{template:LDABOOK|8|The Weibull Distribution}} The Weibull distribution is one of the most widely used lifetime distributions in reliability engineering. It is a versatile distribution that can take on the characteristics of other types of distributions, based on the value of the shape parameter, <math> {\beta} \,\!</math>. This chapter provides a brief background on the Weibull distribution, presents and derives most of the applicable equations and presents example..."

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Lisa Hacker: Reverted edits by Lisa Hacker (talk) to last revision by M Spivey

2023-03-08T00:07:37Z

Reverted edits by Lisa Hacker (talk) to last revision by M Spivey

← Older revision		Revision as of 00:07, 8 March 2023
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	{{template:LDABOOK\|8\|The Weibull Distribution}}		{{template:LDABOOK\|8\|The Weibull Distribution}}
			The Weibull distribution is one of the most widely used lifetime distributions in reliability engineering. It is a versatile distribution that can take on the characteristics of other types of distributions, based on the value of the shape parameter, <math> {\beta} \,\!</math>. This chapter provides a brief background on the Weibull distribution, presents and derives most of the applicable equations and presents examples calculated both manually and by using ReliaSoft's [https://koi-3QN72QORVC.marketingautomation.services/net/m?md=Rw01CJDOxn%2FabhkPlZsy6DwBQ%2BaCXsGR Weibull++ software].

			== Weibull Probability Density Function ==
			===The 3-Parameter Weibull===
			{{three-parameter weibull distribution}}

			===The 2-Parameter Weibull ===
			The 2-parameter Weibull ''pdf'' is obtained by setting
			<math> \gamma=0 \,\!</math>, and is given by:

			:<math> f(t)={ \frac{\beta }{\eta }}\left( {\frac{t}{\eta }}\right) ^{\beta -1}e^{-\left( { \frac{t}{\eta }}\right) ^{\beta }} \,\!</math>

			=== The 1-Parameter Weibull===
			The 1-parameter Weibull ''pdf'' is obtained by again setting
			<math>\gamma=0 \,\!</math> and assuming <math>\beta=C=Constant \,\!</math> assumed value or:

			::<math> f(t)={ \frac{C}{\eta }}\left( {\frac{t}{\eta }}\right) ^{C-1}e^{-\left( {\frac{t}{ \eta }}\right) ^{C}} \,\!</math>

			where the only unknown parameter is the scale parameter, <math>\eta\,\!</math>.

			Note that in the formulation of the 1-parameter Weibull, we assume that the shape parameter <math>\beta \,\!</math> is known ''a priori'' from past experience with identical or similar products. The advantage of doing this is that data sets with few or no failures can be analyzed.

			==Weibull Distribution Functions==
			{{:Weibull Distribution Functions}}

			== Characteristics of the Weibull Distribution ==
			{{:Weibull Distribution Characteristics}}

			== Estimation of the Weibull Parameters ==
			The estimates of the parameters of the Weibull distribution can be found graphically via probability plotting paper, or analytically, using either least squares (rank regression) or maximum likelihood estimation (MLE).

	=== Probability Plotting ===		=== Probability Plotting ===

Lisa Hacker at 00:01, 8 March 2023

2023-03-08T00:01:54Z

← Older revision		Revision as of 00:01, 8 March 2023
Line 1:		Line 1:
	{{template:LDABOOK\|8\|The Weibull Distribution}}		{{template:LDABOOK\|8\|The Weibull Distribution}}
	The Weibull distribution is one of the most widely used lifetime distributions in reliability engineering. It is a versatile distribution that can take on the characteristics of other types of distributions, based on the value of the shape parameter, <math> {\beta} \,\!</math>. This chapter provides a brief background on the Weibull distribution, presents and derives most of the applicable equations and presents examples calculated both manually and by using ReliaSoft's [https://koi-3QN72QORVC.marketingautomation.services/net/m?md=Rw01CJDOxn%2FabhkPlZsy6DwBQ%2BaCXsGR Weibull++ software].

	~~== Weibull Probability Density Function ==~~
	~~===The 3-Parameter Weibull===~~
	~~{{three-parameter weibull distribution}}~~

	~~===The 2-Parameter Weibull ===~~
	~~The 2-parameter Weibull ''pdf'' is obtained by setting~~
	~~<math> \gamma=0 \,\!</math>, and is given by:~~

	~~:<math> f(t)={ \frac{\beta }{\eta }}\left( {\frac{t}{\eta }}\right) ^{\beta -1}e^{-\left( { \frac{t}{\eta }}\right) ^{\beta }} \,\!</math>~~

	~~=== The 1-Parameter Weibull===~~
	~~The 1-parameter Weibull ''pdf'' is obtained by again setting~~
	~~<math>\gamma=0 \,\!</math> and assuming <math>\beta=C=Constant \,\!</math> assumed value or:~~

	~~::<math> f(t)={ \frac{C}{\eta }}\left( {\frac{t}{\eta }}\right) ^{C-1}e^{-\left( {\frac{t}{ \eta }}\right) ^{C}} \,\!</math>~~

	~~where the only unknown parameter is the scale parameter, <math>\eta\,\!</math>.~~

	Note that in the formulation of the 1-parameter Weibull, we assume that the shape parameter <math>\beta \,\!</math> is known ''a priori'' from past experience with identical or similar products. The advantage of doing this is that data sets with few or no failures can be analyzed.

	~~==Weibull Distribution Functions==~~
	~~{{:Weibull Distribution Functions}}~~

	~~== Characteristics of the Weibull Distribution ==~~
	~~{{:Weibull Distribution Characteristics}}~~


	~~== Estimation of the Weibull Parameters ==~~
	The estimates of the parameters of the Weibull distribution can be found graphically via probability plotting paper, or analytically, using either least squares (rank regression) or maximum likelihood estimation (MLE).

	=== Probability Plotting ===		=== Probability Plotting ===

Lisa Hacker at 23:53, 7 March 2023

2023-03-07T23:53:18Z

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Lisa Hacker at 23:51, 7 March 2023

2023-03-07T23:51:20Z

← Older revision		Revision as of 23:51, 7 March 2023
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	Note that in the formulation of the 1-parameter Weibull, we assume that the shape parameter <math>\beta \,\!</math> is known ''a priori'' from past experience with identical or similar products. The advantage of doing this is that data sets with few or no failures can be analyzed.		Note that in the formulation of the 1-parameter Weibull, we assume that the shape parameter <math>\beta \,\!</math> is known ''a priori'' from past experience with identical or similar products. The advantage of doing this is that data sets with few or no failures can be analyzed.

	==Weibull Distribution Functions==		==Weibull Distribution Functions==
	{{:Weibull Distribution Functions}}		{{:Weibull Distribution Functions}}

			== Characteristics of the Weibull Distribution ==
			{{:Weibull Distribution Characteristics}}

Lisa Hacker at 23:50, 7 March 2023

2023-03-07T23:50:36Z

← Older revision		Revision as of 23:50, 7 March 2023
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	==Weibull Distribution Functions==		==Weibull Distribution Functions==
	{{:Weibull Distribution Functions}}		{{:Weibull Distribution Functions}}

	~~== Characteristics of the Weibull Distribution ==~~
	~~{{:Weibull Distribution Characteristics}}~~