5.3.3 Lognormal Distribution
The lognormal distribution is the distribution of a random
variable whose natural logarithm is distributed normally; in other words, it
is the normal distribution with ln
t as the variate. The density function
is


for t ³ 0 
(5.30) 
where the mean = exp 

(5.31) 
and the standard deviation
= 

(5.32) 
where m and s are the mean and standard deviation (SD) of
ln (t).
The lognormal distribution is used
in reliability analysis of semiconductors and fatigue life of certain types of
mechanical components. This distribution is also commonly used in
maintainability analysis and will be further discussed in Section
5.6.2.1.
The cumulative distribution
function for the lognormal is,


(5.33) 
this can be related to the standard
normal variant Z by


(5.34) 
the reliability function is
1F(t) or


(5.35) 
the hazard function, h(t), is given as follows


(5.36) 
where Æ is the standard normal
probability function and m and s are the mean and t standard deviation of the natural
logarithm of the random variable t.