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| {{template:LDABOOK|13|The Gamma Distribution}} | | {{template:LDABOOK|13|The Gamma Distribution}} |
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| ==The Gamma Distribution==
| | {{gamma weibull 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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| {{gamma probability density function}}
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| {{gamma reliability function}}
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| {{gamma mean median and mode}} | |
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| {{gamma standard deviation}}
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| {{gamma reliable life}}
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| {{gamma failure rate function}}
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| {{characteristics of the gamma distribution}}
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| {{gd confidence bounds}}
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| ===Bounds on Time===
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| The bounds around time for a given gamma percentile (unreliability) are estimated by first solving the reliability equation with respect to time, as follows:
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| ::<math>\widehat{T}(\widehat{\mu },\widehat{\sigma })=\widehat{\mu }+\widehat{\sigma }z</math>
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| :where:
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| ::<math>z=\ln (-\ln (R))</math>
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| ::<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>
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| :or:
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| ::<math>Var(\widehat{T})=Var(\widehat{\mu })+2\widehat{z}Cov(\widehat{\mu },\widehat{\sigma })+{{\widehat{z}}^{2}}Var(\widehat{\sigma })</math>
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| The upper and lower bounds are then found by:
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| ::<math>\begin{align}
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| & {{T}_{U}}= & \hat{T}+{{K}_{\alpha }}\sqrt{Var(\hat{T})}\text{ (Upper bound)} \\
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| & {{T}_{L}}= & \hat{T}-{{K}_{\alpha }}\sqrt{Var(\hat{T})}\text{ (Lower bound)}
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| \end{align}</math>
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| ====A Gamma Distribution Example====
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| Twenty four units were reliability tested and the following life test data were obtained:
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| <center><math>\begin{matrix}
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| \text{61} & \text{50} & \text{67} & \text{49} & \text{53} & \text{62} \\
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| \text{53} & \text{61} & \text{43} & \text{65} & \text{53} & \text{56} \\
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| \text{62} & \text{56} & \text{58} & \text{55} & \text{58} & \text{48} \\
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| \text{66} & \text{44} & \text{48} & \text{58} & \text{43} & \text{40} \\
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| \end{matrix}</math></center>
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| Fitting the gamma distribution to this data, using maximum likelihood as the analysis method, gives the following parameters:
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| ::<math>\begin{align}
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| & \hat{\mu }= & 7.72E-02 \\
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| & \hat{k}= & 50.4908
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| \end{align}</math>
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| Using rank regression on <math>X,</math> the estimated parameters are:
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| ::<math>\begin{align}
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| & \hat{\mu }= & 0.2915 \\
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| & \hat{k}= & 41.1726
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| \end{align}</math>
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| Using rank regression on <math>Y,</math> the estimated parameters are:
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| ::<math>\begin{align}
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| & \hat{\mu }= & 0.2915 \\
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| & \hat{k}= & 41.1726
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| \end{align}</math>
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