Template:Gamma distribution - Revision history

Lisa Hacker: Redirected page to The Gamma Distribution

2012-08-03T08:50:47Z

Redirected page to The Gamma Distribution

← Older revision		Revision as of 08:50, 3 August 2012
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	~~{{gamma distribuiton introduction}}~~		#REDIRECT [[The_Gamma_Distribution]]

	~~{{gamma probability density function}}~~

	~~{{gamma reliability function}}~~

	~~{{gamma mean median and mode}}~~

	~~{{gamma standard deviation}}~~

	~~{{gamma reliable life}}~~

	~~{{gamma failure rate function}}~~

	~~{{characteristics of the gamma distribution}}~~

	~~{{Gamma distribution confidence bounds}}~~

	~~===General Example===~~
	~~'''Example 1:'''~~
	~~{{Example: Gamma Distribution Example}}~~

Harry Guo at 22:56, 14 February 2012

2012-02-14T22:56:58Z

← Older revision		Revision as of 22:56, 14 February 2012
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	===General Example===		===General Example===
	'''Example 1:'''		'''Example 1:'''
	{{~~gamma distribution example~~}}		{{Example: Gamma Distribution Example}}

Harry Guo at 22:25, 14 February 2012

2012-02-14T22:25:46Z

← Older revision		Revision as of 22:25, 14 February 2012
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	{{characteristics of the gamma distribution}}		{{characteristics of the gamma distribution}}

	{{gd confidence bounds}}		{{Gamma distribution confidence bounds}}

	===General Example===		===General Example===
	'''Example 1:'''		'''Example 1:'''
	{{gamma distribution example}}		{{gamma distribution example}}

Harry Guo: moved Template:Gamma weibull distribution to Template:Gamma distribution

2012-02-14T22:18:31Z

moved Template:Gamma weibull distribution to Template:Gamma distribution

← Older revision	Revision as of 22:18, 14 February 2012
(No difference)

Harry Guo at 22:11, 14 February 2012

2012-02-14T22:11:23Z

← Older revision		Revision as of 22:11, 14 February 2012
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	{{gd confidence bounds}}		{{gd confidence bounds}}

			===General Example===
			'''Example 1:'''
	{{gamma distribution example}}		{{gamma distribution example}}

Harry Guo at 22:07, 14 February 2012

2012-02-14T22:07:35Z

← Older revision		Revision as of 22:07, 14 February 2012
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	{{gd confidence bounds}}		{{gd confidence bounds}}

	~~====Bounds on Time====~~
	~~The bounds around time for a given gamma percentile (unreliability) are estimated by first solving the reliability equation with respect to time, as follows:~~

	~~::<math>\widehat{T}(\widehat{\mu },\widehat{\sigma })=\widehat{\mu }+\widehat{\sigma }z</math>~~

	~~:where:~~

	~~::<math>z=\ln (-\ln (R))</math>~~

	::<math>Var(\widehat{T})={{(\frac{\partial T}{\partial \mu })}^{2}}Var(\widehat{\mu })+2(\frac{\partial T}{\partial \mu })(\frac{\partial T}{\partial \sigma })Cov(\widehat{\mu },\widehat{\sigma })+{{(\frac{\partial T}{\partial \sigma })}^{2}}Var(\widehat{\sigma })</math>

	~~:or:~~

	~~::<math>Var(\widehat{T})=Var(\widehat{\mu })+2\widehat{z}Cov(\widehat{\mu },\widehat{\sigma })+{{\widehat{z}}^{2}}Var(\widehat{\sigma })</math>~~

	~~The upper and lower bounds are then found by:~~

	~~::<math>\begin{align}~~
	~~& {{T}_{U}}= & \hat{T}+{{K}_{\alpha }}\sqrt{Var(\hat{T})}\text{ (Upper bound)} \\~~
	~~& {{T}_{L}}= & \hat{T}-{{K}_{\alpha }}\sqrt{Var(\hat{T})}\text{ (Lower bound)}~~
	~~\end{align}</math>~~

	{{gamma distribution example}}		{{gamma distribution example}}

Harry Guo: /* Bounds on Time */

2012-02-14T22:03:16Z

Bounds on Time

← Older revision		Revision as of 22:03, 14 February 2012
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	{{gd confidence bounds}}		{{gd confidence bounds}}

	===Bounds on Time===		====Bounds on Time====
	The bounds around time for a given gamma percentile (unreliability) are estimated by first solving the reliability equation with respect to time, as follows:		The bounds around time for a given gamma percentile (unreliability) are estimated by first solving the reliability equation with respect to time, as follows:

Harry Guo at 22:01, 14 February 2012

2012-02-14T22:01:37Z

← Older revision		Revision as of 22:01, 14 February 2012
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	~~==The Gamma Distribution==~~		{{gamma distribuiton introduction}}
	~~The~~ gamma distribution is a flexible life distribution model that may offer a good fit to some sets of failure data. It is not, however, widely used as a life distribution model for common failure mechanisms. The gamma distribution does arise naturally as the time-to-first-fail distribution for a system with standby exponentially distributed backups, and is also a good fit for the sum of independent exponential random variables. The gamma distribution is sometimes called the Erlang distribution, which is used frequently in queuing theory applications. [32]

	{{gamma probability density function}}		{{gamma probability density function}}

Nicolette Young: /* A Gamma Distribution Example */

2012-01-19T22:24:46Z

A Gamma Distribution Example

← Older revision		Revision as of 22:24, 19 January 2012
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	\end{align}</math>		\end{align}</math>

	~~====A Gamma Distribution Example====~~		{{gamma distribution example}}
	~~Twenty four units were reliability tested and the following life test data were obtained:~~

	~~<center><math>\begin~~{~~matrix}~~
	~~\text{61} & \text{50} & \text{67} & \text{49} & \text{53} & \text{62} \\~~
	~~\text{53} & \text{61} & \text{43} & \text{65} & \text{53} & \text{56} \\~~
	~~\text{62} & \text{56} & \text{58} & \text{55} & \text{58} & \text{48} \\~~
	~~\text{66} & \text{44} & \text{48} & \text{58} & \text{43} & \text~~{~~40} \\~~
	~~\end{matrix}</math></center>~~

	~~Fitting the~~ gamma distribution ~~to this data, using maximum likelihood as the analysis method, gives the following parameters:~~

	~~::<math>\begin{align}~~
	~~& \hat{\mu }= & 7.72E-02 \\~~
	~~& \hat{k}= & 50.4908~~
	~~\end{align}</math>~~

	~~Using rank regression on <math>X,</math> the estimated parameters are:~~

	~~::<math>\begin{align}~~
	~~& \hat{\mu }= & 0.2915 \\~~
	~~& \hat{k}= & 41.1726~~
	~~\end{align~~}~~</math>~~


	~~Using rank regression on <math>Y,</math> the estimated parameters are:~~

	~~::<math>\begin{align}~~
	~~& \hat{\mu }= & 0.2915 \\~~
	~~& \hat{k}= & 41.1726~~
	~~\end{align~~}~~</math>~~

Nicolette Young: Created page with '==The Gamma Distribution== The gamma distribution is a flexible life distribution model that may offer a good fit to some sets of failure data. It is not, however, widely used as…'

2012-01-16T22:40:11Z

Created page with '==The Gamma Distribution== The gamma distribution is a flexible life distribution model that may offer a good fit to some sets of failure data. It is not, however, widely used as…'

New page

==The Gamma Distribution==
The gamma distribution is a flexible life distribution model that may offer a good fit to some sets of failure data. It is not, however, widely used as a life distribution model for common failure mechanisms. The gamma distribution does arise naturally as the time-to-first-fail distribution for a system with standby exponentially distributed backups, and is also a good fit for the sum of independent exponential random variables. The gamma distribution is sometimes called the Erlang distribution, which is used frequently in queuing theory applications. [32]

{{gamma probability density function}}

{{gamma reliability function}}

{{gamma mean median and mode}}

{{gamma standard deviation}}

{{gamma reliable life}}

{{gamma failure rate function}}

{{characteristics of the gamma distribution}}

{{gd confidence bounds}}

===Bounds on Time===
The bounds around time for a given gamma percentile (unreliability) are estimated by first solving the reliability equation with respect to time, as follows:

::<math>\widehat{T}(\widehat{\mu },\widehat{\sigma })=\widehat{\mu }+\widehat{\sigma }z</math>

:where:

::<math>z=\ln (-\ln (R))</math>

::<math>Var(\widehat{T})={{(\frac{\partial T}{\partial \mu })}^{2}}Var(\widehat{\mu })+2(\frac{\partial T}{\partial \mu })(\frac{\partial T}{\partial \sigma })Cov(\widehat{\mu },\widehat{\sigma })+{{(\frac{\partial T}{\partial \sigma })}^{2}}Var(\widehat{\sigma })</math>

:or:

::<math>Var(\widehat{T})=Var(\widehat{\mu })+2\widehat{z}Cov(\widehat{\mu },\widehat{\sigma })+{{\widehat{z}}^{2}}Var(\widehat{\sigma })</math>

The upper and lower bounds are then found by:

::<math>\begin{align}
& {{T}_{U}}= & \hat{T}+{{K}_{\alpha }}\sqrt{Var(\hat{T})}\text{ (Upper bound)} \\
& {{T}_{L}}= & \hat{T}-{{K}_{\alpha }}\sqrt{Var(\hat{T})}\text{ (Lower bound)}
\end{align}</math>

====A Gamma Distribution Example====
Twenty four units were reliability tested and the following life test data were obtained:

<center><math>\begin{matrix}
\text{61} & \text{50} & \text{67} & \text{49} & \text{53} & \text{62} \\
\text{53} & \text{61} & \text{43} & \text{65} & \text{53} & \text{56} \\
\text{62} & \text{56} & \text{58} & \text{55} & \text{58} & \text{48} \\
\text{66} & \text{44} & \text{48} & \text{58} & \text{43} & \text{40} \\
\end{matrix}</math></center>

Fitting the gamma distribution to this data, using maximum likelihood as the analysis method, gives the following parameters:

::<math>\begin{align}
& \hat{\mu }= & 7.72E-02 \\
& \hat{k}= & 50.4908
\end{align}</math>

Using rank regression on <math>X,</math> the estimated parameters are:

::<math>\begin{align}
& \hat{\mu }= & 0.2915 \\
& \hat{k}= & 41.1726
\end{align}</math>

Using rank regression on <math>Y,</math> the estimated parameters are:

::<math>\begin{align}
& \hat{\mu }= & 0.2915 \\
& \hat{k}= & 41.1726
\end{align}</math>