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]
This section includes the following subsections:
The pdf of the gamma distribution is given by:
where:
(10)
and:
eμ = scale parameter
k = shape parameter
where 0 <
t < 
, - < μ < and k > 0.
The reliability for a mission of time T for the gamma distribution is:
The gamma mean or MTTF is:
The mode exists if k > 1 and is given by:
The median is:
The standard deviation for the gamma distribution is:
The gamma reliable life is:
The instantaneous gamma failure rate is given by:
Some of the specific characteristics of the gamma distribution are the following:
For k > 1:
As 
,
f(T) increases from 0 to the mode value and decreases thereafter.
If k
2 then pdf
has one inflection point at 
If k >
2 then pdf has
two inflection points for 
For a fixed k, as μ increases, the pdf starts to look more like a straight angle.
As 
For k = 1:
Gamma becomes the exponential distribution.
As 
,
As 
The pdf decreases monotonically and is convex.
. λ(T) is constant.
The mode does not exist.
For 0 < k < 1:
As ,
As
As 
The pdf decreases monotonically and is convex.
As μ increases, the pdf gets stretched out to the right and its height decreases, while maintaining its shape.
As μ decreases, the pdf shifts towards the left and its height increases.
The mode does not exist.
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 in the Confidence Bounds chapter.
The lower and upper bounds on the mean, 
, are estimated
from:
Since the standard deviation, 
, must be positive, ln()
is treated as normally distributed and the bounds are estimated from:
where Kα is defined by:
If δ
is the confidence level, then 
for the two-sided bounds
and α = 1 - δ
for the one-sided bounds.
The variances and covariances of and 
are
estimated from the Fisher matrix, as follows:
Λ is the log-likelihood function of the gamma distribution, described in the Statistical Background chapter and Appendix C.
The reliability of the gamma distribution is:
where:
The upper and lower bounds on reliability are:
(11)
(12)
where:
(13)
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:
Twenty four 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 X, the estimated parameters are:
Using rank regression on Y, the estimated parameters are:
See Also:
Other Distributions
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