Random Variation in a Model

Pearson Type VI

The density function of the Pearson Type VI distribution is

\[ f(x) = \frac{\left(\frac{x}{\beta }\right)^{\alpha _1 - 1}}{\beta G(\alpha _1, \alpha _2)\left[1 + \left(\frac{x}{\beta }\right)\right]^{\alpha _1 + \alpha _2}} \]

where $x > 0$ and

\[ G(\alpha _1, \alpha _2) = \frac{\Gamma (\alpha _1)\Gamma (\alpha _2)}{\Gamma (\alpha _1 + \alpha _2)} \]

The function $\Gamma (z)$ is defined in the section Beta.

Parameters:

$\alpha _1$

is a shape parameter, $\alpha _1 > 0$.

$\alpha _2$

is a shape parameter, $\alpha _2 > 0$.

$\beta $

is a scale parameter, $\beta > 0$.

If $X_1$ and $X_2$ are independent random variables with $X_1 \sim \mbox{Gamma}(\alpha _1, \beta )$ and $X_2 \sim \mbox{Gamma}(\alpha _2, 1)$, then $Y = \frac{X_1}{X_2} \sim \mbox{PearsonTypeVI}(\alpha _1, \alpha _2, \beta )$.

Table B.17 shows how the Pearson Type VI distribution parameter names are specified in Simulation Studio (specifically, in the Numeric Source block). The Pearson Type VI distribution is not available with the Distribution option in JMP.

Table B.17: Pearson Type VI Distribution Parameter Names

 

Simulation Studio

JMP

$\alpha _1$

Shape 1

–

$\alpha _2$

Shape 2

–

$\beta $

Scale

–


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 == Pearson Type VI;Shape 2 == 4;Shape 1 == 0.5;Scale == 1

  • Shape 2 == 2