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, as discussed in [32].

Gamma Probability Density Function

The pdf of the gamma distribution is given by:

where:

and:

where , and .

Gamma Distribution Functions

Mean, Median and Mode

The gamma mean or MTTF is:

The mode exists if and is given by:

The median is:

Standard Deviation

The standard deviation for the gamma distribution is:

The Gamma Reliability Function

The reliability for a mission of time 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: