8.1. Introduction
8.1.6. What are the basic lifetime distribution models used for non-repairable populations?
8.1.6.5. |
Gamma |
There are two ways of writing (parameterizing) the gamma distribution
that are common in the literature. In addition, different authors use different
symbols for the shape and scale parameters. Below we show three ways of
writing the gamma, with a = 
= 
, the "shape"
parameter, and b =1/
,
the scale parameter. The first choice of parameters (a,b) will be
the most convenient for later applications of the gamma. EXCEL uses
while Dataplot uses
.
![PDF: f(t,a,b) = (b**a/GAMMA(a))*t**(a-1)*EXP(-b*t) or f(t,alpha,beta) = (1/(beta**alpha*GAMMA(alpha)))*t**(alpha-1)*EXP(-t/beta) or f(t,gamma,beta) = (1/(beta**gamma*GAMMA(gamma)))*t**(gamma-1)*EXP(-t/beta)CDF: F(t) = INTEGRAL[0 to t]f(t)dtRELIABILITY: R(t) = 1 - F(t)FAILURE RATE: h(t) = f(t)/R(t)MEAN: a/b or alpha*beta or gamma*betaVARIANCE: a/b**2 or alpha*beta**2 or gamma*beta**2](eqns/gamform.gif)
.
Another well-known statistical distribution, the Chi-Square, is also
a special case of the gamma. A Chi-Square distribution with n degrees
of freedom is the same as a gamma with a = n/2 and b
= .5 (or
= 2).
The following plots give examples of gamma PDF, CDF and failure rate shapes.
- The gamma 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 does arise naturally as the time-to-first fail distribution for
a system with standby exponentially distributed
backups. If there are n-1 standby backup units and the system and
all backups have exponential lifetimes with parameter ,
then the total lifetime has a gamma distribution with a = n
and b =.
Note:
when a is a positive integer, the gamma is sometimes called an Erlang
distribution. The Erlang distribution is used frequently in queuing
theory applications.
-
A common use of the gamma model occurs in Bayesian
reliability applications. When a system follows an HPP
(exponential) model with a constant repair rate ,
and it is desired to make use of prior information about possible values
of , a gamma
Bayesian prior for
is a convenient and popular choice.
To calculate the PDF, CDF, Reliability and failure rate at time t
for a gamma with parameters a and b = 1/,
use the following Dataplot statements:
LET PDF = GAMPDF(t,a,0,b)Using an example solved in the section on standby models, if a = 2, b = 1/30 and t = 24 months, the statements are:
LET CDF = GAMCDF(t,a,0,b)
LET REL = 1-CDF
LET FR = PDF/REL
| LET PDF = GAMPDF(24, 2, 0, 30) | response is .01198 |
| LET CDF = GAMCDF(24, 2, 0, 30) | response is .1912 |
| LET REL = 1-CDF | response is .8088 |
| LET FR=PDF/REL | response is .0148 |
To generate random gamma data we first have to set the "a" parameter (called "gamma" by Dataplot). The following commands generate 100 gamma data points chosen randomly from a gamma distribution with parameters a and b:
LET GAMMA = aFor the above example this becomes
LET DATA = GAMMA RANDOM NUMBERS FOR I = 1 1 100
LET DATA = (1/b)*DATA
LET GAMMA = 2Continuing this example, we can now do a gamma probability plot of the 100 points in DATA. The commands are
LET DATA = GAMMA RANDOM NUMBERS FOR I = 1 1 100
LET DATA = 30*DATA
LET GAMMA = 2The resulting plot is shown below.
X1LABEL EXPECTED (NORMALIZED) VALUES
Y1LABEL TIME
GAMMA PROBABILITY PLOT DATA
Note that the value of gamma can be estimated using a PPCC plot.
EXCEL also has built-in functions to evaluate the gamma pdf and cdf. The syntax is:
=GAMMADIST(t,a,1/b,FALSE) for the PDF
=GAMMADIST(t,a,1/b,TRUE) for the CDF
