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==The Gamma Distribution==
#REDIRECT [[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>

Latest revision as of 08:50, 3 August 2012

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