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{{gd confidence bounds}}
{{gd confidence bounds}}
===Bounds on Reliability===
The reliability of the gamma distribution is:
::<math>\widehat{R}(T;\hat{\mu },\hat{k})=1-{{\Gamma }_{1}}(\widehat{k};{{e}^{\widehat{z}}})</math>
:where:
::<math>\widehat{z}=\ln (t)-\widehat{\mu }</math>
The upper and lower bounds on reliability are:
::<math>{{R}_{U}}=\frac{\widehat{R}}{\widehat{R}+(1-\widehat{R})\exp (\tfrac{-{{K}_{\alpha }}\sqrt{Var(\widehat{R})\text{ }}}{\widehat{R}(1-\widehat{R})})}\text{  (upper bound)}</math>
::<math>{{R}_{L}}=\frac{\widehat{R}}{\widehat{R}+(1-\widehat{R})\exp (\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{R})\text{ }}}{\widehat{R}(1-\widehat{R})})}\text{  (lower bound)}</math>
:where:
::<math>Var(\widehat{R})={{(\frac{\partial R}{\partial \mu })}^{2}}Var(\widehat{\mu })+2(\frac{\partial R}{\partial \mu })(\frac{\partial R}{\partial k})Cov(\widehat{\mu },\widehat{k})+{{(\frac{\partial z}{\partial k})}^{2}}Var(\widehat{k})</math>


===Bounds on Time===
===Bounds on Time===

Revision as of 21:26, 4 January 2012


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]

New format available! This reference is now available in a new format that offers faster page load, improved display for calculations and images, more targeted search and the latest content available as a PDF. As of September 2023, this Reliawiki page will not continue to be updated. Please update all links and bookmarks to the latest reference at help.reliasoft.com/reference/life_data_analysis

Chapter 13: The Gamma Distribution


Weibullbox.png

Chapter 13  
The Gamma Distribution  

Synthesis-icon.png

Available Software:
Weibull++

Examples icon.png

More Resources:
Weibull++ Examples Collection


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]

Template loop detected: Template:Gamma probability density function

Template loop detected: Template:Gamma reliability function

Template loop detected: Template:Gamma mean median and mode

Template loop detected: Template:Gamma standard deviation

Template loop detected: Template:Gamma reliable life

Template loop detected: Template:Gamma failure rate function

Template loop detected: Template:Characteristics of the gamma distribution

Template loop detected: Template: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]\displaystyle{ \widehat{T}(\widehat{\mu },\widehat{\sigma })=\widehat{\mu }+\widehat{\sigma }z }[/math]
where:
[math]\displaystyle{ z=\ln (-\ln (R)) }[/math]
[math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ \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:

[math]\displaystyle{ \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]

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

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 7.72E-02 \\ & \hat{k}= & 50.4908 \end{align} }[/math]

Using rank regression on [math]\displaystyle{ X, }[/math] the estimated parameters are:

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 0.2915 \\ & \hat{k}= & 41.1726 \end{align} }[/math]


Using rank regression on [math]\displaystyle{ Y, }[/math] the estimated parameters are:

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 0.2915 \\ & \hat{k}= & 41.1726 \end{align} }[/math]

New format available! This reference is now available in a new format that offers faster page load, improved display for calculations and images, more targeted search and the latest content available as a PDF. As of September 2023, this Reliawiki page will not continue to be updated. Please update all links and bookmarks to the latest reference at help.reliasoft.com/reference/life_data_analysis

Chapter 13: The Gamma Distribution


Weibullbox.png

Chapter 13  
The Gamma Distribution  

Synthesis-icon.png

Available Software:
Weibull++

Examples icon.png

More Resources:
Weibull++ Examples Collection


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]

Template loop detected: Template:Gamma probability density function

Template loop detected: Template:Gamma reliability function

Template loop detected: Template:Gamma mean median and mode

Template loop detected: Template:Gamma standard deviation

Template loop detected: Template:Gamma reliable life

Template loop detected: Template:Gamma failure rate function

Template loop detected: Template:Characteristics of the gamma distribution

Template loop detected: Template: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]\displaystyle{ \widehat{T}(\widehat{\mu },\widehat{\sigma })=\widehat{\mu }+\widehat{\sigma }z }[/math]
where:
[math]\displaystyle{ z=\ln (-\ln (R)) }[/math]
[math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ \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:

[math]\displaystyle{ \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]

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

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 7.72E-02 \\ & \hat{k}= & 50.4908 \end{align} }[/math]

Using rank regression on [math]\displaystyle{ X, }[/math] the estimated parameters are:

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 0.2915 \\ & \hat{k}= & 41.1726 \end{align} }[/math]


Using rank regression on [math]\displaystyle{ Y, }[/math] the estimated parameters are:

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 0.2915 \\ & \hat{k}= & 41.1726 \end{align} }[/math]

New format available! This reference is now available in a new format that offers faster page load, improved display for calculations and images, more targeted search and the latest content available as a PDF. As of September 2023, this Reliawiki page will not continue to be updated. Please update all links and bookmarks to the latest reference at help.reliasoft.com/reference/life_data_analysis

Chapter 13: The Gamma Distribution


Weibullbox.png

Chapter 13  
The Gamma Distribution  

Synthesis-icon.png

Available Software:
Weibull++

Examples icon.png

More Resources:
Weibull++ Examples Collection


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]

Template loop detected: Template:Gamma probability density function

Template loop detected: Template:Gamma reliability function

Template loop detected: Template:Gamma mean median and mode

Template loop detected: Template:Gamma standard deviation

Template loop detected: Template:Gamma reliable life

Template loop detected: Template:Gamma failure rate function

Template loop detected: Template:Characteristics of the gamma distribution

Template loop detected: Template: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]\displaystyle{ \widehat{T}(\widehat{\mu },\widehat{\sigma })=\widehat{\mu }+\widehat{\sigma }z }[/math]
where:
[math]\displaystyle{ z=\ln (-\ln (R)) }[/math]
[math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ \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:

[math]\displaystyle{ \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]

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

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 7.72E-02 \\ & \hat{k}= & 50.4908 \end{align} }[/math]

Using rank regression on [math]\displaystyle{ X, }[/math] the estimated parameters are:

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 0.2915 \\ & \hat{k}= & 41.1726 \end{align} }[/math]


Using rank regression on [math]\displaystyle{ Y, }[/math] the estimated parameters are:

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 0.2915 \\ & \hat{k}= & 41.1726 \end{align} }[/math]

New format available! This reference is now available in a new format that offers faster page load, improved display for calculations and images, more targeted search and the latest content available as a PDF. As of September 2023, this Reliawiki page will not continue to be updated. Please update all links and bookmarks to the latest reference at help.reliasoft.com/reference/life_data_analysis

Chapter 13: The Gamma Distribution


Weibullbox.png

Chapter 13  
The Gamma Distribution  

Synthesis-icon.png

Available Software:
Weibull++

Examples icon.png

More Resources:
Weibull++ Examples Collection


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]

Template loop detected: Template:Gamma probability density function

Template loop detected: Template:Gamma reliability function

Template loop detected: Template:Gamma mean median and mode

Template loop detected: Template:Gamma standard deviation

Template loop detected: Template:Gamma reliable life

Template loop detected: Template:Gamma failure rate function

Template loop detected: Template:Characteristics of the gamma distribution

Template loop detected: Template: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]\displaystyle{ \widehat{T}(\widehat{\mu },\widehat{\sigma })=\widehat{\mu }+\widehat{\sigma }z }[/math]
where:
[math]\displaystyle{ z=\ln (-\ln (R)) }[/math]
[math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ \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:

[math]\displaystyle{ \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]

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

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 7.72E-02 \\ & \hat{k}= & 50.4908 \end{align} }[/math]

Using rank regression on [math]\displaystyle{ X, }[/math] the estimated parameters are:

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 0.2915 \\ & \hat{k}= & 41.1726 \end{align} }[/math]


Using rank regression on [math]\displaystyle{ Y, }[/math] the estimated parameters are:

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 0.2915 \\ & \hat{k}= & 41.1726 \end{align} }[/math]

New format available! This reference is now available in a new format that offers faster page load, improved display for calculations and images, more targeted search and the latest content available as a PDF. As of September 2023, this Reliawiki page will not continue to be updated. Please update all links and bookmarks to the latest reference at help.reliasoft.com/reference/life_data_analysis

Chapter 13: The Gamma Distribution


Weibullbox.png

Chapter 13  
The Gamma Distribution  

Synthesis-icon.png

Available Software:
Weibull++

Examples icon.png

More Resources:
Weibull++ Examples Collection


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]

Template loop detected: Template:Gamma probability density function

Template loop detected: Template:Gamma reliability function

Template loop detected: Template:Gamma mean median and mode

Template loop detected: Template:Gamma standard deviation

Template loop detected: Template:Gamma reliable life

Template loop detected: Template:Gamma failure rate function

Template loop detected: Template:Characteristics of the gamma distribution

Template loop detected: Template: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]\displaystyle{ \widehat{T}(\widehat{\mu },\widehat{\sigma })=\widehat{\mu }+\widehat{\sigma }z }[/math]
where:
[math]\displaystyle{ z=\ln (-\ln (R)) }[/math]
[math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ \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:

[math]\displaystyle{ \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]

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

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 7.72E-02 \\ & \hat{k}= & 50.4908 \end{align} }[/math]

Using rank regression on [math]\displaystyle{ X, }[/math] the estimated parameters are:

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 0.2915 \\ & \hat{k}= & 41.1726 \end{align} }[/math]


Using rank regression on [math]\displaystyle{ Y, }[/math] the estimated parameters are:

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 0.2915 \\ & \hat{k}= & 41.1726 \end{align} }[/math]

New format available! This reference is now available in a new format that offers faster page load, improved display for calculations and images, more targeted search and the latest content available as a PDF. As of September 2023, this Reliawiki page will not continue to be updated. Please update all links and bookmarks to the latest reference at help.reliasoft.com/reference/life_data_analysis

Chapter 13: The Gamma Distribution


Weibullbox.png

Chapter 13  
The Gamma Distribution  

Synthesis-icon.png

Available Software:
Weibull++

Examples icon.png

More Resources:
Weibull++ Examples Collection


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]

Template loop detected: Template:Gamma probability density function

Template loop detected: Template:Gamma reliability function

Template loop detected: Template:Gamma mean median and mode

Template loop detected: Template:Gamma standard deviation

Template loop detected: Template:Gamma reliable life

Template loop detected: Template:Gamma failure rate function

Template loop detected: Template:Characteristics of the gamma distribution

Template loop detected: Template: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]\displaystyle{ \widehat{T}(\widehat{\mu },\widehat{\sigma })=\widehat{\mu }+\widehat{\sigma }z }[/math]
where:
[math]\displaystyle{ z=\ln (-\ln (R)) }[/math]
[math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ \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:

[math]\displaystyle{ \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]

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

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 7.72E-02 \\ & \hat{k}= & 50.4908 \end{align} }[/math]

Using rank regression on [math]\displaystyle{ X, }[/math] the estimated parameters are:

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 0.2915 \\ & \hat{k}= & 41.1726 \end{align} }[/math]


Using rank regression on [math]\displaystyle{ Y, }[/math] the estimated parameters are:

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 0.2915 \\ & \hat{k}= & 41.1726 \end{align} }[/math]

New format available! This reference is now available in a new format that offers faster page load, improved display for calculations and images, more targeted search and the latest content available as a PDF. As of September 2023, this Reliawiki page will not continue to be updated. Please update all links and bookmarks to the latest reference at help.reliasoft.com/reference/life_data_analysis

Chapter 13: The Gamma Distribution


Weibullbox.png

Chapter 13  
The Gamma Distribution  

Synthesis-icon.png

Available Software:
Weibull++

Examples icon.png

More Resources:
Weibull++ Examples Collection


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]

Template loop detected: Template:Gamma probability density function

Template loop detected: Template:Gamma reliability function

Template loop detected: Template:Gamma mean median and mode

Template loop detected: Template:Gamma standard deviation

Template loop detected: Template:Gamma reliable life

Template loop detected: Template:Gamma failure rate function

Template loop detected: Template:Characteristics of the gamma distribution

Template loop detected: Template: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]\displaystyle{ \widehat{T}(\widehat{\mu },\widehat{\sigma })=\widehat{\mu }+\widehat{\sigma }z }[/math]
where:
[math]\displaystyle{ z=\ln (-\ln (R)) }[/math]
[math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ \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:

[math]\displaystyle{ \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]

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

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 7.72E-02 \\ & \hat{k}= & 50.4908 \end{align} }[/math]

Using rank regression on [math]\displaystyle{ X, }[/math] the estimated parameters are:

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 0.2915 \\ & \hat{k}= & 41.1726 \end{align} }[/math]


Using rank regression on [math]\displaystyle{ Y, }[/math] the estimated parameters are:

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 0.2915 \\ & \hat{k}= & 41.1726 \end{align} }[/math]

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Chapter 13: The Gamma Distribution


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Chapter 13  
The Gamma Distribution  

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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]

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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]\displaystyle{ \widehat{T}(\widehat{\mu },\widehat{\sigma })=\widehat{\mu }+\widehat{\sigma }z }[/math]
where:
[math]\displaystyle{ z=\ln (-\ln (R)) }[/math]
[math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ \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:

[math]\displaystyle{ \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]

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

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 7.72E-02 \\ & \hat{k}= & 50.4908 \end{align} }[/math]

Using rank regression on [math]\displaystyle{ X, }[/math] the estimated parameters are:

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 0.2915 \\ & \hat{k}= & 41.1726 \end{align} }[/math]


Using rank regression on [math]\displaystyle{ Y, }[/math] the estimated parameters are:

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 0.2915 \\ & \hat{k}= & 41.1726 \end{align} }[/math]

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]\displaystyle{ \widehat{T}(\widehat{\mu },\widehat{\sigma })=\widehat{\mu }+\widehat{\sigma }z }[/math]
where:
[math]\displaystyle{ z=\ln (-\ln (R)) }[/math]
[math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ \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:

[math]\displaystyle{ \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]

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

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 7.72E-02 \\ & \hat{k}= & 50.4908 \end{align} }[/math]

Using rank regression on [math]\displaystyle{ X, }[/math] the estimated parameters are:

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 0.2915 \\ & \hat{k}= & 41.1726 \end{align} }[/math]


Using rank regression on [math]\displaystyle{ Y, }[/math] the estimated parameters are:

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 0.2915 \\ & \hat{k}= & 41.1726 \end{align} }[/math]