Random Variation in a Model

Johnson Bounded Distribution (JohnsonSB)

The density function of the Johnson bounded distribution (JohnsonSB) is

\[ f(x) = \frac{\delta }{\lambda \sqrt {2\pi }}g’\left(\frac{x - \xi }{\lambda }\right)\mbox{exp}\left(-\frac{1}{2}\left[\gamma + \delta g\left(\frac{x - \xi }{\lambda }\right)\right]^2\right) \]

where

\[ \begin{array}{l} g(y) = \mbox{ln}\left(\frac{y}{1 - y}\right)\\ g’(y) = \frac{1}{y(1 - y)} \end{array} \]

and $x \in [\xi , \xi + \lambda ]$.

Parameters:

$\delta $ (delta)

is a shape parameter, $\delta > 0$.

$\gamma $ (gamma)

is a shape parameter.

$\xi $ (xi)

is the location parameter.

$\lambda $ (lambda)

is the scale parameter, $\lambda > 0$.

Table B.11 shows how the Johnson bounded distribution parameter names are specified in Simulation Studio (specifically, in the Numeric Source block) and in JMP.

Table B.11: Johnson Bounded Distribution Parameter Names

 

Simulation Studio

JMP

$\gamma $

gamma

$\gamma $

$\delta $

delta

$\delta $

$\xi $

xi

$\theta $

$\lambda $

lambda

$\sigma $


The following examples show (case-sensitive) string values that can be used as Numeric Source block DataStreamDescription factor values or InStreamPolicy port values. In these examples, the distribution and parameter names in the string value are the names that are used in the Theoretical option in the Numeric Source Block Properties dialog box (including any spaces or hyphens). Quotation marks are not required around the string value, and you can specify only the parameters that need to be updated (as demonstrated in the second example).

  • class == Johnson Bounded;gamma == 0;delta == 2;xi == 0;lambda == 1

  • gamma == 0.5;delta == 0.8