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probabilistic Design analysis • 85
3.2.3.4 Parameter estimation

The method of moments estimators for A and B are:

A = x − 3 S

B = x + 3 S

The maximum likelihood estimators for A and B are:

Y
A = midrang YY … ) − 05 rang YY … )
.
,
,
,
,
Y ,
,
n
n
2
2
( 1
( 1
B = midrang YY … ) + 05 rang YY … )
Y
,
Y ,
,
,
,
,
.
n
n
2
( 1
( 1
2
3.2.3.5 Comments
The uniform distribution is a very fundamental distribution for cases
where no other information apart from a lower and an upper limit
exists. It is very useful to describe geometric tolerances. It can also be
used in cases where there is no evidence that any value of the random
variable is more likely than any other within a certain interval. In this
sense, it can be used for cases where “lack of engineering knowledge”
plays a role.
The uniform distribution defines equal probability over a given range for
a continuous distribution. For this reason, it is important as a reference dis-
tribution. One of the most important applications of the uniform distribution
is in the generation of random numbers. That is, almost all random number
generators generate random numbers on the (0, 1) interval. For other dis-
tributions, some transformation is applied to the uniform random numbers.
3.2.4 LognoRMAL DiSTRibUTion
A variable X is lognormally distributed if Y = LN(X) is normally distrib-
uted with “LN” denoting the natural logarithm. The general formula for
the pdf of the lognormal distribution is:
− ( ( ln x− ) q / m)) 2
( )
(
e / 2 s 2
m
fx () = � x ≥ q;,s > 0
( x − ) qs 2 p