| Meaning: |
-
Distribution of \(T = exp(Z)\), if \(Z\) is normally distributed with the
parameters \(\mu\) and \(\sigma^2\).
-
Limit distribution of the product of many independent random variables with
arbitrary statistical characteristics if the contribution of a single random
variable remains very small.
-
In its form similar to a gamma or Weibull distribution-
|
| Parameters: |
-
\(\mu\) (mean value of \(Z\))
-
\(\sigma > 0\) (standart deviation of \(Z\))
Alternative parameters mean value and coefficient of variation:
-
\(\mu = ln(\frac{E[T]}{\sqrt{1+c_T^2}})\)
-
\(\sigma = \sqrt{ln(1+c_T^2)}\)
|
| PDF: |
\(P(T=t) = f(t) = \frac{1}{\sqrt{2\pi} \sigma \cdot t} \cdot exp(
-\frac{(ln t -\mu)^2}{2 \sigma^2} ) \mbox{ for } t>0 \) |
| DF: |
No closed form |
| Expected value: |
\(E[T]= exp(\mu + \frac{\sigma^2}{2})\) |
| Variance: |
\(VAR[T]= exp(2 \mu + \sigma^2) \cdot (exp(\sigma^2) -1)\) |
| Coefficient of variation: |
\(c_T= \sqrt{exp(\sigma^2) -1}\) |
| Parser example: |
[...].Distribution = Lognormal
[...].Distribution.My = 0.11
[...].Distribution.Sigma = 1.27
or with the mean value and variation coefficient:
[...].Distribution.Mean = 2.5
[...].Distribution.CoefficientOfVariation = 2.0
|