---
title: "Poisson random walk"
canonical: "https://modelassist.epixanalytics.com/space/EA/26575445/Poisson%20random%20walk"
format: markdown
---
The [Poisson](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575214/) random walk assumes the following equation:

 

S<sub>t</sub>=Poisson(m*t+c)

 

The model [Poisson Random Walk](#Poisson_random_walk) can be used, for example, to describe vehicle accident claims made to an insurance company, or cases of a disease for a health authority: as the number of cars increases, the number of car crashes increases correspondingly according to some function; as the pollution level in a city increases, the number of people with respiratory disease increases.

 

 

![image](media://4ceb14f6-84b6-4dd9-a98b-618001bdfea0)

 

The fractional variation of the series is much bigger on the left panel than that on the right panel. This is because the standard deviation of Poisson counts [equals √](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575986/)<span style="color: #222222">[λ](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575986/)</span>. Thus, the [coefficient of variance](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575507/) (std.dev./mean) is 1/√<span style="color: #222222">λ</span>*.* which gets smaller as <span style="color: #222222">λ</span> gets bigger, meaning that the larger the expected number of events, the smaller the fractional variation one would observe. This property of a [Poisson process](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575359) is very useful to insurance companies: the more people they cover, the more stable their liabilities become, and the less margin they need to cover themselves at a certain risk level. An example of when big is actually better.

> Macro (anchor)



The links to the Poisson Random Walk software specific models are provided here:

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

> Macro (inline-media-image)

 [Poisson_random_walk](/wiki/download/attachments/26579343/Poisson_random_walk-CB.xlsx?version=1&modificationDate=1506113500000&cacheVersion=1&api=v2)

 

Here is a screenshot of the model:

![image593.gif](media://8da29272-009c-4d19-8749-3d5be0a4a2c7)
</details>

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

> Macro (inline-media-image)

 [Poisson_random_walk](/wiki/download/attachments/26581760/Poisson_random_walk-AtRISK.xlsx?version=1&modificationDate=1506097520000&cacheVersion=1&api=v2)

 

Here is a screenshot of the model:

![image593.gif](media://8da29272-009c-4d19-8749-3d5be0a4a2c7)
</details>

 

 

 

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