Introduction

• A very handy and easy to use calculator, supported with a powerful visual plot was designed to help with quick probality calculations of one of the most importatnt distributions in Statistics with a wide range of real life apllications…..The Poisson Distribution.

• By following a few simple instructions one will be able to quickly compute poisson probabilities for a wide range of scenarios.

The Poisson Distribution

• Arguably the second most important distribution after the Normal distribution.
• A Poisson random variable is a discrete variable which can be described as a count of the number of occurrences of an event, where events occur randomly and independently at a constant rate(lambda).
• If these conditions are satisfied, then X, the number of events in a fixed unit interval (T), has a Poisson distribution.
• The interval can be any measure, not necessarily time depending on the study in question and examples include:

1. Number of customers arriving at a checkout point in an hour (time)
2. Number of defects in a km of a pipeline (distance)
3. Number of patches in a square meter (area)

• As an example: Suppose the mean number of calls to a fire station on a weekday is 8. Then the constant rate lambda = 8 and the fixed unit interval T = 1 day and our X will be the number of calls. Decision makers are usually interested in the probabilities of a certain number of events occurring within a certain interval. In our example above, one might ask:

1. What is the probability that on a given weekday there would be 11 calls?
2. What is the probability that on a given weekday there would be 11 or less calls?
3. What is the probability of getting 11 calls in half a day?

Answering the Questions

• These questions can be answered by hand by substituting into the probability density function formula for the Poisson distribution

\[p(x|\lambda) = \frac{\lambda^xe^{-\lambda}}{x!}\] where x = 0,1,2,3,….

• This can be fairly straight forward if one is only interested in the probability of exactly x occurances (11 in our example) in the specified unit interval. This can be computed in R as follows:

dpois(11,8)

## [1] 0.07219021

• However, for the second question above for example, then one needs to sum up the probabilities of all the possible values of x 11 or less which would require repeating the above calculation 12 times!

• Our equation becomes:

\[p(0<= x <= 11|\lambda) =\sum_{x=0}^{11} \frac{\lambda^xe^{-\lambda}}{x!}\]

• Yet in R it can be done quickly :

sum(dpois(0:11,8))

## [1] 0.888076

The App

• Very simple to use!
• Just follow a set of simple instructions:

1. Enter lambda (the average number of occurances per given interval)
2. Enter the number of occurances the probability of which you would like to compute
3. Enter proprtion of interval you would like to investigate. For example if your lambda is x occurances per day and you want to investigate half a day, then enter 0.5. If you want to investigate 2 days, then enter 2. Note that this will adjust your lambda accordingly.
4. Submit…Probabilities are displayed on the caculator tab and the visualisation is dispalyed on the plot tab.
5. You can adjust your visualisation by by adjusting the number of points to plot to suit your investigation, and press submit.

• With a little imagination a person with an undestanding of probabilities will be able to handle most variations including differnt ranges of occurences with minimal calculations.

• Here is a link to the App:

browseURL("https://gombevincent.shinyapps.io/poissondistribution/")

Links to App and Codes

• The app can be accessed via this url: https://gombevincent.shinyapps.io/poissondistribution

• The server and UI codes are saved in this GitHub repo:

https://github.com/vincentgombe/Developing-Data-Products-Project.git