The lognormal distribution is commonly used for general reliability analysis, cycles-to-failure in fatigue, material strengths and loading variables in probabilistic design. A random variable is lognormally distributed if the logarithm of the random variable is normally distributed. Since the logarithms of a lognormally distributed random variable are normally distributed, the lognormal distribution is given by:
where:
= ln T,
and where the Ts
are the times-to-failure.
= mean
of the natural logarithms of the times to failure.
= standard
deviation of the natural logarithms of the times to failure.
The lognormal pdf can be obtained, realizing that for equal probabilities under the normal and lognormal pdfs incremental areas should also be equal, or:
Taking the derivative yields:
Substitution yields:
where:
Statistical Properties Summary
The Mean or MTTF
The mean of the lognormal distribution, 
, is given by:
(12)
The mean of the natural logarithms of the times-to-failure, , in terms of and 
is given
by:
The Standard Deviation
The standard deviation of the lognormal distribution,
, is given by:
(13)
The standard deviation of the natural logarithms of the times-to-failure,
, in terms of and is given
by:
The Median
The median of the lognormal distribution is given by:
(14)
The Mode
The mode of the lognormal distribution is given by:
Reliability Function
For the lognormal distribution, the reliability for a mission of time T, starting at age 0, is given by:
or:
There is no closed form solution for the lognormal reliability function. Solutions can be obtained via the use of standard normal tables.
Lognormal Failure Rate
The lognormal failure rate is given by:
This subchapter includes the following topics:
See Also:
Life Distributions
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