The Weibull distribution is one of the most commonly used distributions in reliability engineering because of the many shapes it attains for various values of β (slope). It can therefore model a great variety of data and life characteristics [ 19].
The 2-parameter Weibull pdf is given by:
where:
and:
η = scale parameter.
β = shape parameter (or slope).
Weibull Statistical Properties Summary
The Mean or MTTF
The mean, 
of the 2-parameter
Weibull pdf is given by:
where 
is the gamma function
evaluated at the value of 
.
The Median
The median, 
, of the 2-parameter
Weibull is given by:
(6)
The Mode
The mode, 
, of the 2-parameter
Weibull is given by:
(7)
The Standard Deviation
The standard deviation, 
of
the 2-parameter Weibull is given by:
The cdf and the Reliability Function
The cdf of the 2-parameter Weibull distribution is given by:
The Weibull reliability function is given by:
The Conditional Reliability Function
The Weibull conditional reliability function is given by:
(8)
or:
Equation (8) gives the reliability for a new mission of t duration, having already accumulated T hours of operation up to the start of this new mission and the units are checked out to assure that they will start the next mission successfully. (It is called conditional because you can calculate the reliability of a new mission based on the fact that the unit(s) already accumulated T hours of operation successfully).
The Reliable Life
For the 2-parameter Weibull distribution, the reliable life, TR, of a unit for a specified reliability, starting the mission at age zero, is given by:
(9)
This is the life for which the unit will function successfully with
a reliability of R(TR).
If R(TR)
= 0.50 then TR
= , the median life, or the life by which
half of the units will survive.
The Failure Rate Function
The 2-parameter Weibull failure rate function, λ(T), is given by:
This subchapter includes the following topics:
See Also:
Life Distributions
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