Template:WeibullDistribution - Revision history

Lisa Hacker: Redirected page to The Weibull Distribution

2012-08-09T10:12:11Z

Redirected page to The Weibull Distribution

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	[[~~Category: For Deletion~~]]		#REDIRECT [[The Weibull Distribution]]

Lisa Hacker: Replaced content with 'Category: For Deletion'

2012-07-20T12:14:15Z

Replaced content with 'Category: For Deletion'

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	~~=== The Weibull Distribution ===~~		[[Category: For Deletion]]

	The Weibull distribution is a general purpose reliability distribution used to model material strength, times-to-failure of electronic and mechanical components, equipment or systems. In its most general case, the three-parameter Weibull <span class="texhtml">''p''''d''''f''</span> is defined by: <br>

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

	~~<br>~~

	with three parameters <span class="texhtml">β</span> , <span class="texhtml">η</span> and <span class="texhtml">γ,</span> where <span class="texhtml">β = </span>shape parameter, <span class="texhtml">η = </span>scale parameter and γ = location parameter. <br>If the location parameter, <span class="texhtml">γ</span> , is assumed to be zero, then the distribution becomes the two-parameter Weibull or:

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

	One additional form is the one-parameter Weibull distribution, which assumes that the location parameter, <span class="texhtml">γ,</span> is zero, and the shape parameter is a known constant, or <span class="texhtml">β = </span>constant <span class="texhtml">= ''C''</span>, so:

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

	~~The Weibull distribution and its characteristics are presented in~~ [[~~The Weibull Distribution~~]].

	~~<br>{{weibull-bayesian distribution}}~~

Lisa Hacker: /* The Weibull Distribution */

2012-03-30T21:41:11Z

The Weibull Distribution

← Older revision		Revision as of 21:41, 30 March 2012
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	</math>		</math>

	The Weibull distribution and its characteristics are presented ~~in detail~~ in [[The Weibull Distribution]].		The Weibull distribution and its characteristics are presented in [[The Weibull Distribution]].

	<br>{{weibull-bayesian distribution}}		<br>{{weibull-bayesian distribution}}

Lisa Hacker: /* The Weibull Distribution */

2012-03-30T21:36:37Z

The Weibull Distribution

← Older revision		Revision as of 21:36, 30 March 2012
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	</math>		</math>

	[[The Weibull Distribution]] ~~fully details the Weibull distribution and presents many examples of its use in Weibull++~~.		The Weibull distribution and its characteristics are presented in detail in [[The Weibull Distribution]].

	<br>{{weibull-bayesian distribution}}		<br>{{weibull-bayesian distribution}}

Lisa Hacker: /* The Weibull Distribution */

2012-03-30T21:28:14Z

The Weibull Distribution

← Older revision		Revision as of 21:28, 30 March 2012
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	</math>		</math>

	~~The chapter~~ [[The Weibull Distribution]] ~~of this reference~~ fully details the Weibull distribution and presents many examples of its use in Weibull++.		[[The Weibull Distribution]] fully details the Weibull distribution and presents many examples of its use in Weibull++.

	<br>{{weibull-bayesian distribution}}		<br>{{weibull-bayesian distribution}}

Kate Racaza at 16:00, 12 March 2012

2012-03-12T16:00:36Z

← Older revision		Revision as of 16:00, 12 March 2012
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	===The Weibull Distribution===		=== The Weibull Distribution ===
	The Weibull distribution is a general purpose reliability distribution used to model material strength, times-to-failure of electronic and mechanical components, equipment or systems. In its most general case, the three-parameter Weibull <~~math~~>~~pdf~~</~~math~~> is defined by:
	<br>		The Weibull distribution is a general purpose reliability distribution used to model material strength, times-to-failure of electronic and mechanical components, equipment or systems. In its most general case, the three-parameter Weibull <span class="texhtml">''p''''d''''f''</span> is defined by: <br>

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

	<br>		<br>

	with three parameters <~~math~~>~~\beta~~ </~~math~~> , <~~math~~>~~\eta~~ </~~math~~> and <~~math~~>~~\gamma~~ ,</~~math~~> where <~~math~~>~~\beta~~ =</~~math~~> shape parameter, <~~math~~>~~\eta~~ =</~~math~~> scale parameter and location parameter.		with three parameters <span class="texhtml">β</span> , <span class="texhtml">η</span> and <span class="texhtml">γ,</span> where <span class="texhtml">β = </span>shape parameter, <span class="texhtml">η = </span>scale parameter and γ = location parameter. <br>If the location parameter, <span class="texhtml">γ</span> , is assumed to be zero, then the distribution becomes the two-parameter Weibull or:
	<br>
	If the location parameter, <~~math~~>~~\gamma~~ </~~math~~> , is assumed to be zero, the distribution ~~then~~ becomes the two-parameter Weibull or:

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

	One additional form is the one-parameter Weibull distribution, which assumes that the location parameter, <~~math~~>~~\gamma~~ ,</~~math~~> is zero, and the shape parameter is a known constant, or <~~math~~>~~\beta~~ =</~~math~~> constant <~~math~~>=C</~~math~~>, so:		One additional form is the one-parameter Weibull distribution, which assumes that the location parameter, <span class="texhtml">γ,</span> is zero, and the shape parameter is a known constant, or <span class="texhtml">β = </span>constant <span class="texhtml">= ''C''</span>, so:

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

	~~Chapter~~ [[The Weibull Distribution]] of this reference fully details the Weibull distribution and presents many examples of its use in Weibull++.		The chapter [[The Weibull Distribution]] of this reference fully details the Weibull distribution and presents many examples of its use in Weibull++.

	<br>		<br>{{weibull-bayesian distribution}}
	{{weibull-bayesian distribution}}

Harry Guo: /* The Weibull Distribution */

2012-02-11T00:01:04Z

The Weibull Distribution

← Older revision		Revision as of 00:01, 11 February 2012
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	</math>		</math>

	[[The Weibull Distribution ~~\| Chapter 8~~]] of this reference fully details the Weibull distribution and presents many examples of its use in Weibull++.		Chapter [[The Weibull Distribution]] of this reference fully details the Weibull distribution and presents many examples of its use in Weibull++.

	<br>		<br>
	{{weibull-bayesian distribution}}		{{weibull-bayesian distribution}}

Harry Guo: /* The Weibull Distribution */

2012-02-03T21:14:35Z

The Weibull Distribution

← Older revision		Revision as of 21:14, 3 February 2012
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	</math>		</math>

	Chapter 6 of this reference fully details the Weibull distribution and presents many examples of its use in Weibull++.		[[The Weibull Distribution \| Chapter 8]] of this reference fully details the Weibull distribution and presents many examples of its use in Weibull++.

	<br>		<br>
	{{weibull-bayesian distribution}}		{{weibull-bayesian distribution}}

Nicolette Young: /* The Weibull-Bayesian Distribution */

2012-01-06T19:16:00Z

The Weibull-Bayesian Distribution

← Older revision		Revision as of 19:16, 6 January 2012
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	<br>		<br>
	~~====The Weibull~~-~~Bayesian Distribution====~~		{{weibull-bayesian distribution}}
	Another approach is the Weibull-Bayesian model which assumes that the analyst has some prior knowledge about the distribution of the shape parameter ( <math>\beta )</math> of the Weibull distribution. There are many practical applications for this model, particularly when dealing with small sample sizes and/or some prior knowledge for the shape parameter is available. For example, when a test is performed, there is often a good understanding about the behavior of the failure mode under investigation, primarily through historical data or physics-of-failure.
	~~<br>~~
	Note that this is not the same as the so called WeiBayes model. The so called WeiBayes model is really a one-parameter Weibull distribution. It assumes a fixed value (constant) for the shape parameter and solves for the scale parameter. The Weibull-Bayesian model in Weibull++ 7 is actually a true WeiBayes model and offers an alternative to the one-parameter Weibull by including the variation and uncertainty that is present in the prior estimation of the shape parameter.
	~~<br>~~
	~~The Weibull-Bayesian~~ distribution ~~and its characteristics are presented in more detail in Chapter 6.~~

Nicolette Young: Created page with '===The Weibull Distribution=== The Weibull distribution is a general purpose reliability distribution used to model material strength, times-to-failure of electronic and mechanic…'

2012-01-03T23:06:36Z

Created page with '===The Weibull Distribution=== The Weibull distribution is a general purpose reliability distribution used to model material strength, times-to-failure of electronic and mechanic…'

New page

===The Weibull Distribution===
The Weibull distribution is a general purpose reliability distribution used to model material strength, times-to-failure of electronic and mechanical components, equipment or systems. In its most general case, the three-parameter Weibull <math>pdf</math> is defined by:
 
::<math>f(t)=\frac{\beta}{\eta } \left( \frac{t-\gamma }{\eta } \right)^{\beta -1}{e}^{-(\tfrac{t-\gamma }{\eta }) ^{\beta}}</math>
 

with three parameters <math>\beta </math> , <math>\eta </math> and <math>\gamma ,</math> where <math>\beta =</math> shape parameter, <math>\eta =</math> scale parameter and location parameter.
 
If the location parameter, <math>\gamma </math> , is assumed to be zero, the distribution then becomes the two-parameter Weibull or:

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

One additional form is the one-parameter Weibull distribution, which assumes that the location parameter, <math>\gamma ,</math> is zero, and the shape parameter is a known constant, or <math>\beta =</math> constant <math>=C</math>, so:

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

Chapter 6 of this reference fully details the Weibull distribution and presents many examples of its use in Weibull++.

 
====The Weibull-Bayesian Distribution====
Another approach is the Weibull-Bayesian model which assumes that the analyst has some prior knowledge about the distribution of the shape parameter ( <math>\beta )</math> of the Weibull distribution. There are many practical applications for this model, particularly when dealing with small sample sizes and/or some prior knowledge for the shape parameter is available. For example, when a test is performed, there is often a good understanding about the behavior of the failure mode under investigation, primarily through historical data or physics-of-failure.
 
Note that this is not the same as the so called WeiBayes model. The so called WeiBayes model is really a one-parameter Weibull distribution. It assumes a fixed value (constant) for the shape parameter and solves for the scale parameter. The Weibull-Bayesian model in Weibull++ 7 is actually a true WeiBayes model and offers an alternative to the one-parameter Weibull by including the variation and uncertainty that is present in the prior estimation of the shape parameter.
 
The Weibull-Bayesian distribution and its characteristics are presented in more detail in Chapter 6.