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The RELIABILITY Procedure |
Note that you should not interpret the parameters and
as representing the means and standard deviations for all of the
distributions in Table 30.37.
The normal is the only distribution in Table 30.37
for which this is the case.
Parameters | |||||
Distribution | Density Function | Location | Scale | Shape | Threshold |
Normal | ![]() | ![]() | ![]() | ||
Lognormal | ![]() | ![]() | ![]() | ![]() | |
Lognormal | ![]() | ![]() | ![]() | ![]() | |
(base 10) | |||||
Extreme Value | ![]() | ![]() | ![]() | ||
Weibull | ![]() | ![]() | ![]() | ![]() | |
Exponential | ![]() | ![]() | ![]() | ||
Logistic | ![]() | ![]() | ![]() | ||
Log-logistic | ![]() | ![]() | ![]() | ![]() | |
The exponential distribution shown in Table 30.37 is a special
case of the Weibull distribution with .The remaining distributions in Table 30.37 are related to one another
as shown in Table 30.38.
The threshold parameter,
, is
assumed to be 0 in Table 30.38.
Distribution of T | Parameters | Distribution of Y=logT | Parameters | ||
Lognormal | ![]() | ![]() | Normal | ![]() | ![]() |
Weibull | ![]() | ![]() | Extreme Value | ![]() | ![]() |
Log-logistic | ![]() | ![]() | Logistic | ![]() | ![]() |
If a lifetime T has the generalized gamma distribution, then the logarithm of the lifetime X = log(T) has the generalized log-gamma distribution, with the following probability density function g(x). When the gamma distribution is specified, the logarithms of the lifetimes are used as responses, and the generalized log-gamma distribution is used to estimate the parameters by maximum likelihood.
When , the generalized log-gamma distribution
reduces to the extreme value
distribution with parameters
and
.In this case, the log lifetimes have the extreme value
distribution, or, equivalently, the lifetimes
have the Weibull distribution with parameters
and
.When
,the generalized log-gamma reduces to the normal distribution with parameters
and
. In this case, the (unlogged) lifetimes have
the lognormal distribution with parameters
and
.This chapter uses the notation
for the location,
for the
scale, and
for the shape parameters for the generalized
log-gamma distribution.
Distribution | Pr{Y=y} | Parameter | Parameter Name |
Binomial | ![]() | p | binomial probability |
Poisson | ![]() | ![]() | Poisson mean |
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