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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]
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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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| {{gamma distribution example}}
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