Probability Distributions > What is the Poisson Distribution?

A Poisson distribution is a tool that helps to predict the probability of certain events from happening when you know how often the event has occurred. The Poisson distribution gives us the **probability of a given number of events happening in a fixed interval of time**.

### Practical Example of the Poisson Distribution

A store that rents books has an average rental of 200 books every Saturday night. Using this data, you can **predict the probability that more books will sell** (perhaps 300 or 400) on the following Saturday nights. Another example is the number of diners in a certain restaurant every day. If the average number of diners for seven days is 500, you can predict the probability of a certain day having more customers.

Because of this application, Poisson distributions are used by businessmen to make **forecasts** about the number of customers or sales on certain days or seasons of the year. In business, overstocking will sometimes mean losses if the goods are not sold. Likewise, having too few stocks would still mean a lost business opportunity because you were not able to maximize your sales due to a shortage of stock. By using this tool, businessmen are able to estimate the time when demand is unusually higher, so they can purchase more stock. Hotels and restaurants could prepare for an influx of customers, they could hire extra temporary workers in advance, purchase more supplies, or make contingency plans just in case they cannot accommodate their guests coming to the area.

With the Poisson distribution, companies can adjust supply to demand in order to keep their business earning good profit. In addition, waste of resources is prevented.

### Calculating the Poisson Distribution

The Poisson Distribution pmf is: P(x; μ) = (e

^{-μ}* μ^{x}) / x!

The symbol “!” is a factorial.

μ (the expected number of occurrences) is sometimes written as λ. Sometimes called the **event rate** or rate parameter.

**Sample question:** The average number of major storms in your city is 2 per year. What is the probability that exactly 3 storms will hit your city next year?

Step 1: Figure out the components you need to put into the equation.

μ = 2 (average number of storms per year, historically)

x = 3 (the number of storms we think might hit next year)

e = 2.71828 (e is a constant)

Step 2: Plug the values from Step 1 into the Poisson distribution formula:

P(x; μ) = (e^{-μ}) (μ^{x}) / x!

= (2.71828 ^{– 2}) (2^{3}) / 3!

= (0.13534) (8) / 6

= 0.180

The probability of 3 storms happening next year is 0.180, or 18%

As you can probably tell, you can calculate the Poisson distribution manually but that would take an extraordinary amount of time unless you have a simple set of data. The usual way to calculate a Poisson distribution in real life situations is with software like IBM SPSS.

## Poisson distribution vs. Binomial

The above example was over-simplified to show you how to work through a problem. However, it can be challenging to figure out if you should use a binomial or a Poisson distribution. If you aren’t given a specific guideline from your instructor, use the following general guideline.

- If your question has an
**average probability**of an event happening per unit (i.e. per unit of time, cycle, event)**and**you want to find probability of a certain number of events happening in a period of time (or number of events), then use the Poisson Distribution. - If you are given an
**exact probability**and you want to find the probability of the event happening a certain number out times out of x (i.e. 10 times out of 100, or 99 times out of 1000), use the Binomial Distribution formula.

**See also:** Prime Numbers and the Poisson Distr.

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Incredible and helpful website!

Just a minor mistake above, I think you meant to say 2 storms per YEAR and not month. I used 24 and got this super low probability haha.

You’re right! Thanks for spotting that; now fixed!

There is a problem with the math in the example. The result of (e – 2) is not .13534 and (2*3) is not 8. Or maybe I’m missing something.

You’re right. Thanks for catching that — it’s fixed!

The formula is used wrongly. It is e^-2 and 2^8. So the previous answers, .13534 and 8 are correct.

Thanks for the correction :)