Lognormal(m, s), Lognormal – Rockwell Automation Arena Basic Edition Users Guide User 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