Pearson5(a,b) = b / Gamma(0,1,a)

The Pearson family of distributions was designed by Pearson between 1890 and 1895. It represents a system whereby for every member the probability density function f(x) satisfies a differential equation:

\frac{1}{p}\frac{dp}{dx}=-\frac{a+x}{c_{0}+c_{1}x+c_{2}x^{2}} |

(1)

where the shape of the distribution depends on the values of the parameters a, c0, c1, and c2. The Pearson Type V corresponds to the case where c0 + c1x + c2x2 is a perfect square (c2=4c0c2). Thus, equation (1) can be rewritten as:

\frac{d \;log \;f(x)}{dx}=-\frac{x+a}{c_{2}(x+c_{1})^{2}} |

Examples of the Pearson Type 5 distribution are given below:

#### Uses

This distribution is very rarely properly used in risk analysis.

### Generation

There are two ways to generate values from the Pearson Type 5 Distribution with Crystal Ball:

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 V. 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 5 Distribution using the following algorithm:

Pearson5(a,b) = b / Gamma(0,1,a)

This method is also illustrated in the model Pearson V.

#### Comments

The Pearson family includes many familiar distributions:

The Normal distribution

Beta, Inverse Beta (=1/Beta), Gamma, and Inverse Gamma (=1/Gamma) distributions which usually have an overall bell-shape but are generally skewed left or right

Student t distributions, which are symmetrical (unskewed) but have longer tails than the Normal distribution

Type II distributions, which are symmetric but have thicker, shorter tails than the Normal distribution. The Uniform distribution is of Type II

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