Pearson Type 6

Pearson6(a1,a2,b) = b*X/(1-X) where X = Beta(a1,a2,1)

Pearson Type 6 equations

The Pearson⁽¹⁾ Type 6 distribution corresponds in the Pearson system to the case when the roots of $//<![CDATA[ \begin{array}{l}c_{0}+c_{1}x+c_{2}x^{2}=0\end{array} //]]>$ are real and of the same sign.

Examples of the Pearson Type 6 distribution are given below:

Uses

At Epix Analytics we don't find much use for this distribution (other than to generate an F distribution). The distribution is very unlikely to reflect any of the processes that the analyst may come across, but it's three parameters (giving it flexibility), sharp peak and long tail make it a possible candidate to be fitted to a very large (so you know the pattern is real) data set that other distributions won't fit well to.

Like the Pearson Type 5 distribution , the Pearson⁽¹⁾ Type 6 distribution hasn't proven to be very useful in risk analysis.

Generation

There are two ways to generate values from the Pearson Type 6 Distribution:

Method 1.

The first method is to construct a General distribution from the equation for the probability density function, as shown in the model Pearson VI. While this method retains the benefits of Latin Hypercube sampling if used, it requires quite a lot of space within your spreadsheet model.

Method 2.

It is easy to generate values from the Pearson Type 6 Distribution using the following algorithm:

This method is also illustrated in the model Pearson VI.

The links to the Pearson VI software specific models are provided here:

⁽¹⁾ Karl Pearson (1857 – 1936)