Rockwell Automation Arena Basic Edition Users Guide - Manual (Page 75)

A • S TATISTICAL D ISTRIBUTIONS
69
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A •
Statis
tical
Dis
tribu
tions
Lognormal(
μ
,
σ
)LOGNORMAL(LogMean, LogStd) or
LOGN(LogMean, LogStd)
Mean LogMean ( μ
l
> 0) and standard deviation LogStd ( σ
l
> 0) of the lognormal random
variable. Both LogMean and LogStd must be specified as strictly positive real numbers.
[0, +
LogMean = μ
1
=
(LogStd) 2 = σ
1
2 =
The lognormal distribution is used in situations in which the quantity is the product of a
large number of random quantities. It is also frequently used to represent task times that
have a distribution skewed to the right. This distribution is related to the normal
distribution as follows. If X has a lognormal (
μ
l
, σ
l
) distribution, then ln(X) has a normal
( μ , σ ) distribution. Note that μ and σ are not the mean and standard deviation of the
Probability
Density
Function
Denote the user-specified input parameters as
LogMean =
μ
l
and LogStd = σ
l
.
Then let
the probability density function can then be
written as
) / ln( 2 2 2
l l l
μ σ μ μ + = and
[ ]
2
/ )
2 2
( ln
l l l
μ μ σ σ + =
f(x) =
) 2 /( ) ) (ln( 2 2
2
1
σ μ
π σ
− − x e
x
for x > 0
0 otherwise
Parameters
Range
∞ )
Mean
e
μ σ 2 2 ⁄ +
Variance
e
2μ σ
2 +
e σ 2 1 – ( )
Applications