The Gamma Distribution

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Chapter 13: The Gamma Distribution



Index

Chapter 13  
The Gamma Distribution  

Contents

Available Software:
Weibull++

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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, as discussed in [32].

The Gamma Probability Density Function

The pdf of the gamma distribution is given by:

where:

and:

where , and .

The Gamma Reliability Function

The reliability for a mission of time for the gamma distribution is:

The Gamma Mean, Median and Mode

The gamma mean or MTTF is:

The mode exists if and is given by:

The median is:

The Gamma Standard Deviation

The standard deviation for the gamma distribution is:

The Gamma Reliable Life

The gamma reliable life is:

The Gamma Failure Rate Function

The instantaneous gamma failure rate is given by:

Characteristics of the Gamma Distribution

Some of the specific characteristics of the gamma distribution are the following:

For  :

• As ,
increases from 0 to the mode value and decreases thereafter.
• If then pdf has one inflection point at
• If then pdf has two inflection points for
• For a fixed , as increases, the pdf starts to look more like a straight angle.
• As

For  :

• Gamma becomes the exponential distribution.
• As ,
• As
• The pdf decreases monotonically and is convex.
. is constant.
• The mode does not exist.

For  :

• As ,
• As
• As
• The pdf decreases monotonically and is convex.
• As increases, the pdf gets stretched out to the right and its height decreases, while maintaining its shape.
• As decreases, the pdf shifts towards the left and its height increases.
• The mode does not exist.

Confidence Bounds

The only method available in Weibull++ for confidence bounds for the gamma distribution is the Fisher matrix, which is described next. The complete derivations were presented in detail (for a general function) in the Confidence Bounds chapter.

Bounds on the Parameters

The lower and upper bounds on the mean, , are estimated from:

Since the standard deviation, , must be positive, is treated as normally distributed and the bounds are estimated from:

where is defined by:

If is the confidence level, then for the two-sided bounds and for the one-sided bounds.

The variances and covariances of and are estimated from the Fisher matrix, as follows:


is the log-likelihood function of the gamma distribution, described in Parameter Estimation and Appendix D

Bounds on Reliability

The reliability of the gamma distribution is:

where:

The upper and lower bounds on reliability are:

where:

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:

where:

or:

The upper and lower bounds are then found by:

General Example

24 units were reliability tested, and the following life test data were obtained:

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

Using rank regression on the estimated parameters are:

Using rank regression on the estimated parameters are:

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