---
title: "Pearson Type 5"
canonical: "https://modelassist.epixanalytics.com/space/EA/26575249/Pearson%20Type%205"
format: markdown
---
Pearson5(a,b) = b / Gamma(0,1,a)

[Pearson Type 5 equations](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26576362/)

 

 

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:


(1) 

![image](media://e7db1ab2-c3e7-44dd-ad5a-1e4f6b108180)

 

> Macro (mathblock)

                                                                                            

 

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:

 

![image](media://5352b709-c14e-43de-a40c-f7018f1b1556)

 

> Macro (mathblock)

 

Examples of the Pearson Type 5 distribution are given below:

 

![image](media://f1499307-761b-4ca7-8a11-76d5e6a3e008)

 

#### 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](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575232/) from the equation for the probability density function, as shown in the model [Pearson V](#Pearson_V). While this method retains the benefits of [Latin Hypercube sampling](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575293/) 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](#Pearson_V).

#### Comments

The Pearson family includes many familiar distributions:

 

- The [Normal](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575246/) distribution
- [Beta](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575217/), Inverse Beta (=1/Beta), [Gamma](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575231/), and Inverse Gamma (=1/Gamma) distributions which usually have an overall bell-shape but are generally skewed left or right
- [Student t](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575254/) 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](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575256/) is of Type II

 

> Macro (anchor)



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

<details>
<summary>Crystal Ball</summary>

> Macro (inline-media-image)

 [Pearson_V](/wiki/download/attachments/26579343/Pearson_V-CB.xlsx?version=1&modificationDate=1506113499000&cacheVersion=1&api=v2)
</details>

<details>
<summary>@Risk</summary>

> Macro (inline-media-image)

 [Pearson_V](/wiki/download/attachments/26581760/Pearson_V-AtRISK.xlsx?version=1&modificationDate=1506097519000&cacheVersion=1&api=v2)
</details>

 

 

 

 

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