---
title: "Introduction - The Poisson Process"
canonical: "https://modelassist.epixanalytics.com/space/EA/26574979/Introduction%20-%20The%20Poisson%20Process"
format: markdown
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# The Poisson Process

 

> Macro (gliffy)


[Exposure 'time'](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574981) *[t](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574981)** **[(Gamma)](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575231)*

[Number of observations ](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574980)*[a](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574980)** **[(Poisson)](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575214)*

[Mean number of events per ](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574982/Estimate+of+the+mean+number+of+events+per+period+lambda)[unit time λ](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574982) *(**[Gamma)](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575231)*


Definition

  


In the [binomial process](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574971/), there are *n* discrete opportunities for an event (a 'success') to occur. In the Poisson process, there is a continuous and constant opportunity for an event to occur. For example, lightning strikes might be considered to occur as a Poisson process during a storm. That would mean that in any small time interval during the storm, there is a certain probability that a lightning strike will occur. In the case of lightning strikes, the continuum of opportunity is time. However, there are other types of exposure. The occurrence of discontinuities in the continuous manufacture of wire could be considered to be a Poisson process where the measure of exposure is, for example, kilometers or tones of wire produced.

  


Examples

  


If Giardia cysts were randomly distributed in a lake, the consumption of cysts by campers drinking the water would be a Poisson process, where the measure of exposure would be the total amount of water consumed.

  


Typographic errors in a book might be Poisson distributed, in which case the measure of exposure could be inches of text, although one could just as easily consider the errors to be binomially distributed with *n* = the number of characters in the book.

  


In a Poisson process, unlike the binomial, one can theoretically have anything between zero and an infinite number of events within a specific amount of 'time', and there is a probability of the event occurring no matter how small a unit of exposure we might consider. In practice, few physical systems will exactly conform to such a set of assumptions, but many systems nevertheless in practice are very well approximated by a Poisson process. In the Giardia cyst example above, assuming a Poisson process would theoretically mean that we could have any number of cysts in a volume of water, no matter how small we made that volume. Obviously, this assumption breaks down when we consider a volume of liquid around the size of a cyst, or smaller, but this is almost never a restriction in practice.

  


The distributions

  


The distributions describing the Poisson and binomial processes are strongly related to each other. In a [binomial process](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574970/), the key descriptive parameter is *p*, the probability of occurrence of an event, which is the same for all trials, so the trials are independent of each other. The key descriptive parameter for the Poisson process is <span style="color: #222222">*λ*</span>, the mean number of events that will occur per unit of exposure, which is also considered to be constant over the total amount of exposure *t*. That means that there is a constant probability per second, for example, of an event occurring, whether or not an event has just occurred, has not occurred for an unexpectedly long time, etc. Such a process is called "memoryless" and both the binomial and Poisson processes have this property.

  


Like *p* for a binomial process, <span style="color: #222222">*λ*</span> is a property of the physical system. For static systems (stochastic processes) *p* and <span style="color: #222222">*λ*</span> are not random variables, but we still need distributions to express the state of our knowledge (uncertainty) about their values.

  


In a Poisson process, we are concerned with the number of events that may occur in a period *t*, the amount of 'time' one will have to wait to observe <span style="color: #222222">α</span> events, and <span style="color: #222222">*λ*</span>, the average number of events that occur, that describes the Poisson process. This section continues by showing how the Poisson distribution, which describes the number of events <span style="color: #222222">α</span> that may occur in a period of exposure *t*, can be derived from the [Binomial](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575204/) distribution as *p* tends to zero and *n* tends to infinity. We will then look at how to determine the variability distribution of the time *t* one will need to wait before observing <span style="color: #222222">α</span> events, which also turns out to be the distribution of uncertainty of the time one must have waited before having observed *a* events. Finally, we will discuss how to determine our state of knowledge (uncertainty) about <span style="color: #222222">*λ*</span> given a set of observed events <span style="color: #222222">α</span> in a period *t*.

  


1. [Deriving the Poisson distribution from the Binomial](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574980/)
2. [”Time” to wait to observe ](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574981/)<span style="color: #222222">*[α](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574981/)*</span>[ events](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574981/)
3. [Estimate of the mean number of events per period, lambda](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574982/)
4. [Estimate of the elapsed period ](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574983)*[t](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574983)*

  


<span style="color: #ffffff">Once you have reviewed the material in this section, you might like to test how much you have learned by taking the self-test quiz:</span>

  


<span style="color: #ffffff">A quiz on The Poisson process: </span>> Macro (take-quiz-button)



  


Summary of results for the Poisson process

  


|  |  |  |
| --- | --- | --- |
| ***Quantity*** | ***Formula*** | ***Notes*** |
| Number of events | <span style="color: #222222">*α*</span> = [Poisson](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575214/)(<span style="color: #222222">*λ*</span>*t*) |  |
| Mean number of events per unit exposure of success | <span style="color: #222222">*λ*</span> = [Gamma](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575231/)(0, 1/t, <span style="color: #222222">*α*</span>)<br>= Gamma(0, <span style="color: #222222">β</span>/(1+<span style="color: #222222">β</span>*t*), a+<span style="color: #222222">α</span>) | Assuming uninformed prior<br>Assuming Gamma(0,<span style="color: #222222">β</span>*,a*) prior |
| Time until observe first event | *t*<sub>*1*</sub> = [Exponential](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575227/)(<span style="color: #222222">*λ*</span>)<br>= Gamma(0, 1/<span style="color: #222222">*λ*</span>*, 1*) |  |
| Time until observe first <span style="color: #222222">*α*</span> events | *t*<sub>*a*</sub> = Gamma(0, 1/<span style="color: #222222">λ</span>*,* <span style="color: #222222">α</span>) |  |
| Time that has elapsed for <span style="color: #222222">*α*</span> events | *t*<sub>*a*</sub> = Gamma(0, 1/<span style="color: #222222">λ</span>*,* <span style="color: #222222">*α*</span>)<br>= Gamma(0, <span style="color: #222222">β</span>/(1+<span style="color: #222222">β</span><span style="color: #222222">*λ*</span>), <span style="color: #222222">*α*</span>+*a*) | Assuming uninformed prior<br>Assuming Gamma(0,<span style="color: #222222">β</span>,*a*) prior |

  


Useful Excel Functions:

  


|  |  |  |
| --- | --- | --- |
| **Use** | **Function** | **Explanation** |
| Poisson probability | =[POISSON](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575570/)(x,<span style="color: #222222">λ</span>, cumulative) | The Poisson probability of:<br><span style="color: #222222">α</span> = x (cumulative = FALSE), or:<br><span style="color: #222222">α</span> <= x (cumulative = TRUE) |
| Gamma probability | =[GAMMADIST](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575570/)(x,<span style="color: #222222">α</span>,<span style="color: #222222">β</span>,cumulative) | The Gamma density for time x (cumulative = FALSE), or probability time<x (cumulative = TRUE) |
| Exponential probability | =[EXPONDIST](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575570/)(x,<span style="color: #222222">λ</span>,cumulative) | The Exponential density for time x (cumulative = FALSE), or probability time<x (cumulative = TRUE) |
| Gamma inverse probability | =GAMMAINV(P,<span style="color: #222222">α</span>,<span style="color: #222222">β</span>) | The time x such that P(time <= x) = P |

  


  


  


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