Template:Alta weibull distribution - Revision history

Richard House: Redirected page to Distributions Used in Accelerated Testing#The Weibull Distribution

2012-08-16T01:42:47Z

Redirected page to Distributions Used in Accelerated Testing#The Weibull Distribution

← Older revision		Revision as of 01:42, 16 August 2012
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	#REDIRECT [[Distributions_Used_in_Accelerated_Testing]]		#REDIRECT [[Distributions_Used_in_Accelerated_Testing#The_Weibull_Distribution]]

Lisa Hacker: Redirected page to Distributions Used in Accelerated Testing

2012-08-09T10:14:10Z

Redirected page to Distributions Used in Accelerated Testing

← Older revision		Revision as of 10:14, 9 August 2012
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	~~== The Weibull Distribution ==~~		#REDIRECT [[Distributions_Used_in_Accelerated_Testing]]

	The Weibull distribution is one of the most commonly used distributions in reliability engineering because of the many shapes it attains for various values of <span class="texhtml">β</span> (slope). It can therefore model a great variety of data and life characteristics [[~~Reference Appendix D: References\|[18]]].~~

	~~The 2-parameter Weibull ''pdf'' is given by:~~

	~~<br>~~

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

	~~<br>~~

	~~where:~~

	~~<br>~~

	~~::<math>f(T)\ge 0,\text{ }T\ge 0,\text{ }\beta >0,\text{ }\eta >0\text{ }</math>~~

	~~<br> and:~~

	~~<br>~~

	~~:*<span class="texhtml">η = </span> scale parameter.~~
	~~:*<span class="texhtml">β = </span> shape parameter (or slope).~~

	~~<br> {{aw statistical properties summary}}~~

	~~{{aw characteristics}}~~

	~~=== Parameter Estimation ===~~

	The estimates of the parameters of the Weibull distribution can be found graphically on probability plotting paper, or analytically using either least squares or maximum likelihood. (Parameter estimation methods are presented in detail in [[Appendix B: Parameter Estimation\|Appendix B]]~~.) <br>~~

	~~{{aw probability plotting}}~~

	~~{{mle alta for ed}}~~

Chris Kahn: /* Parameter Estimation */

2012-03-06T23:52:05Z

Parameter Estimation

← Older revision		Revision as of 23:52, 6 March 2012
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	=== Parameter Estimation ===		=== Parameter Estimation ===

	The estimates of the parameters of the Weibull distribution can be found graphically on probability plotting paper, or analytically using either least squares or maximum likelihood. (Parameter estimation methods are presented in detail in [[Appendix B: Parameter Estimation\|Appendix B]].) <br> {{aw probability plotting}}		The estimates of the parameters of the Weibull distribution can be found graphically on probability plotting paper, or analytically using either least squares or maximum likelihood. (Parameter estimation methods are presented in detail in [[Appendix B: Parameter Estimation\|Appendix B]].) <br>

			{{aw probability plotting}}

	{{mle alta for ed}}		{{mle alta for ed}}

Chris Kahn at 23:15, 6 March 2012

2012-03-06T23:15:00Z

← Older revision		Revision as of 23:15, 6 March 2012
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	==The Weibull Distribution==		== The Weibull Distribution ==
	~~<br>~~

	The Weibull distribution is one of the most commonly used distributions in reliability engineering because of the many shapes it attains for various values of <~~math~~>~~\beta~~ </~~math~~> (slope). It can therefore model a great variety of data and life characteristics [[Reference Appendix D: References\|[18]]].		The Weibull distribution is one of the most commonly used distributions in reliability engineering because of the many shapes it attains for various values of <span class="texhtml">β</span> (slope). It can therefore model a great variety of data and life characteristics [[Reference Appendix D: References\|[18]]].

	~~<br>~~		The 2-parameter Weibull ''pdf'' is given by:
	The 2-parameter Weibull ~~<math>~~pdf~~</math>~~ is given by:
			<br>

	~~<br>~~
	::<math>f(T)=\frac{\beta }{\eta }{{\left( \frac{T}{\eta } \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}</math>		::<math>f(T)=\frac{\beta }{\eta }{{\left( \frac{T}{\eta } \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}</math>

	where:		<br>

			where:

			<br>

	::<math>f(T)\ge 0,\text{ }T\ge 0,\text{ }\beta >0,\text{ }\eta >0\text{ }</math>		::<math>f(T)\ge 0,\text{ }T\ge 0,\text{ }\beta >0,\text{ }\eta >0\text{ }</math>
	~~<br>~~
	~~and:~~
	~~:• <math>\eta =</math> scale parameter.~~
	~~:• <math>\beta =</math> shape parameter (or slope).~~
	~~<br>~~
	~~{{aw statistical properties summary}}~~

	{{aw characteristics}}		<br> and:

			<br>

			:*<span class="texhtml">η = </span> scale parameter.
			:*<span class="texhtml">β = </span> shape parameter (or slope).

			<br> {{aw statistical properties summary}}

			{{aw characteristics}}

			=== Parameter Estimation ===

	~~===Parameter Estimation===~~		The estimates of the parameters of the Weibull distribution can be found graphically on probability plotting paper, or analytically using either least squares or maximum likelihood. (Parameter estimation methods are presented in detail in [[Appendix B: Parameter Estimation\|Appendix B]].) <br> {{aw probability plotting}}
	The estimates of the parameters of the Weibull distribution can be found graphically on probability plotting paper, or analytically using either least squares or maximum likelihood. (Parameter estimation methods are presented in detail in [[Appendix B: Parameter Estimation\|Appendix B]] .)
	<br>
	{{aw probability plotting}}

	{{mle alta for ed}}		{{mle alta for ed}}

Athanasios Gerokostopoulos: /* The Weibull Distribution */

2012-02-27T22:31:53Z

The Weibull Distribution

← Older revision		Revision as of 22:31, 27 February 2012
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	<br>		<br>

	The Weibull distribution is one of the most commonly used distributions in reliability engineering because of the many shapes it attains for various values of <math>\beta </math> (slope). It can therefore model a great variety of data and life characteristics [18].		The Weibull distribution is one of the most commonly used distributions in reliability engineering because of the many shapes it attains for various values of <math>\beta </math> (slope). It can therefore model a great variety of data and life characteristics [[Reference Appendix D: References\|[18]]].

	<br>		<br>
	The 2-parameter Weibull <math>pdf</math> is given by:		The 2-parameter Weibull <math>pdf</math> is given by:

	<br>		<br>
	::<math>f(T)=\frac{\beta }{\eta }{{\left( \frac{T}{\eta } \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}</math>		::<math>f(T)=\frac{\beta }{\eta }{{\left( \frac{T}{\eta } \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}</math>

	:where:		where:

	::<math>f(T)\ge 0,\text{ }T\ge 0,\text{ }\beta >0,\text{ }\eta >0\text{ }</math>		::<math>f(T)\ge 0,\text{ }T\ge 0,\text{ }\beta >0,\text{ }\eta >0\text{ }</math>
	<br>		<br>
	:and:		and:
	:• <math>\eta =</math> scale parameter.		:• <math>\eta =</math> scale parameter.
	:• <math>\beta =</math> shape parameter (or slope).		:• <math>\beta =</math> shape parameter (or slope).
Line 21:		Line 23:

	===Parameter Estimation===		===Parameter Estimation===
	The estimates of the parameters of the Weibull distribution can be found graphically on probability plotting paper, or analytically using either least squares or maximum likelihood. (Parameter estimation methods are presented in detail in Appendix B.)		The estimates of the parameters of the Weibull distribution can be found graphically on probability plotting paper, or analytically using either least squares or maximum likelihood. (Parameter estimation methods are presented in detail in [[Appendix B: Parameter Estimation\|Appendix B]] .)
	<br>		<br>
	{{aw probability plotting}}		{{aw probability plotting}}

	{{mle alta for ed}}		{{mle alta for ed}}

Athanasios Gerokostopoulos: /* The Weibull Distribution */

2012-02-13T23:02:07Z

The Weibull Distribution

← Older revision		Revision as of 23:02, 13 February 2012
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	==The Weibull Distribution==		==The Weibull Distribution==
			<br>

	The Weibull distribution is one of the most commonly used distributions in reliability engineering because of the many shapes it attains for various values of <math>\beta </math> (slope). It can therefore model a great variety of data and life characteristics [18].		The Weibull distribution is one of the most commonly used distributions in reliability engineering because of the many shapes it attains for various values of <math>\beta </math> (slope). It can therefore model a great variety of data and life characteristics [18].
	<br>		<br>

Athanasios Gerokostopoulos: /* Parameter Estimation */

2012-02-07T00:29:03Z

Parameter Estimation

@@ Line 22: / Line 22: @@
 <br>
 {{aw probability plotting}}
 {{mle alta for ed}}

Nicolette Young: /* Probability Plotting */

2012-01-12T18:46:24Z

Probability Plotting

← Older revision		Revision as of 18:46, 12 January 2012
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	The estimates of the parameters of the Weibull distribution can be found graphically on probability plotting paper, or analytically using either least squares or maximum likelihood. (Parameter estimation methods are presented in detail in Appendix B.)		The estimates of the parameters of the Weibull distribution can be found graphically on probability plotting paper, or analytically using either least squares or maximum likelihood. (Parameter estimation methods are presented in detail in Appendix B.)
	<br>		<br>
	~~====Probability Plotting====~~		{{aw probability plotting}}
	~~One method of calculating the parameters of the Weibull distribution is by using~~ probability plotting~~. To better illustrate this procedure, consider the following example [18].~~

	~~<br>~~
	====Example 3====		====Example 3====
	Let's assume six identical units are being reliability tested at the same application and operation stress levels. All of these units fail during the test after operating the following times (in hours), <math>{{T}_{i}}</math> : 93, 34, 16, 120, 53 and 75.		Let's assume six identical units are being reliability tested at the same application and operation stress levels. All of these units fail during the test after operating the following times (in hours), <math>{{T}_{i}}</math> : 93, 34, 16, 120, 53 and 75.

Nicolette Young: /* Characteristics */

2012-01-12T18:45:45Z

Characteristics

@@ Line 16: / Line 16: @@
 {{aw statistical properties summary}}
-===Characteristics===
+{{aw characteristics}}
 ===Parameter Estimation===

Nicolette Young: /* Statistical Properties Summary */

2012-01-12T18:17:52Z

Statistical Properties Summary

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 :•	 <math>\beta =</math>  shape parameter (or slope).
 <br>
-===Statistical Properties Summary===
+{{aw statistical properties summary}}
 ===Characteristics===