## What is the Poisson Distribution?

The Poisson Distribution is a tool used in probability theory statistics to predict the amount of variation from a known average rate of occurrence, within a given time frame.

In other words, if the average rate at which a specific event happens within a specified time frame is known or can be determined (e.g., Event “A” happens, on average, “x” times per hour), then the Poisson Distribution can be used as follows:

- To determine how much variation there will likely be from that average number of occurrences
- To determine the probable maximum and minimum number of times the event will occur within the specified time frame

Companies can utilize the Poisson Distribution to examine how they may be able to take steps to improve their operational efficiency. For instance, an analysis done with the Poisson Distribution might reveal how a company can arrange staffing in order to be able to better handle peak periods for customer service calls.

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### The History of the Poisson Distribution

Like many statistical tools and probability metrics, the Poisson Distribution was originally applied to the world of gambling. In 1830, French mathematician Siméon Denis Poisson developed the distribution to indicate the low to high spread of the probable number of times that a gambler would win at a gambling game – such as baccarat – within a large number of times that the game was played. (Unfortunately, the gambler paid no heed to Poisson’s prediction of the probabilities of his obtaining only a certain number of wins, and lost heavily.)

The wide range of possible applications of Poisson’s statistical tool became evident several years later, during World War II, when a British statistician used it to analyze bomb hits in the city of London. R.D. Clarke refined the Poisson Distribution as a statistical model and worked to reassure the British government that the German bombs fell randomly, or purely by chance, and that its enemies lacked sufficient information to be targeting certain areas of the city.

Since then, the Poisson Distribution’s been applied across a wide range of fields of study, including medicine, astronomy, business, and sports.

### When the Poisson Distribution is Valid

The Poisson Distribution is only a valid probability analysis tool under certain conditions. It is a valid statistical model if all the following conditions exist:

*k*is the number of times an event happens within a specified time period, and the possible values for*k*are simple numbers such as 0, 1, 2, 3, 4, 5, etc.- No occurrence of the event being analyzed affects the probability of the event re-occurring (events occur independently).
- The event in question cannot occur twice at exactly the same time; there must be some interval of time – even if just half a second – that separates occurrences of the event.
- The probability of an event happening within a portion of the total time frame being examined is proportional to the length of that smaller portion of the time frame.

- The number of trials (chances for the event to occur) is sufficiently greater than the number of times the event does actually occur (in other words, the Poisson Distribution is only designed to be applied to events that occur relatively rarely)

Given the above conditions, then *k* is a random variable, and the distribution of *k* is a Poisson Distribution.

### The Distribution Formula

Below is the Poisson Distribution formula, where the mean (average) number of events within a specified time frame is designated by μ. The probability formula is:

##### P(*x*; μ) = (e^{-μ}) (μ^{x}) / x!

Where:

*x* = number of times and event occurs during the time period

*e* (Euler’s number = the base of natural logarithms) is approx. 2.72

*x!* = the factorial of x (for example is x is 3 then x! = 3 x 2 x 1 = 6)

Let’s see the formula in action:

Say that on average the daily sales volume of 60-inch 4K-UHD TVs at XYZ Electronics is five. Calculate the probability of XYZ Electronics selling nine TVs today.

- μ = 5, since five 60-inch TVs is the daily sales average
- x = 9, because we want to solve for the probability of nine TVs being sold
- e = 2.71828

Insert the values into the distribution formula: P(*x*; μ) = (e^{-μ}) (μ^{x}) / x!

= (2.71828^{-5}) (5^{9}) / 9!

= (0.0067) (1953125) / (3262880)

= 0.036

**3.6%** is the probability of nine 60-inch TVs being sold today.

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### Examples: Business Uses of the Poisson Distribution

The Poisson Distribution can be practically applied to several business operations that are common for companies to engage in. As noted above, analyzing operations with the Poisson Distribution can provide company management with insights into levels of operational efficiency and suggest ways to increase efficiency and improve operations.

Here are some of the ways that a company might utilize analysis with the Poisson Distribution.

- Check for adequate customer service staffing. Calculate the average number of customer service calls per hour that require more than 10 minutes to handle. Then, calculate the Poisson Distribution to find the probable maximum number of calls per hour that might come in requiring more than ten minutes to handle. Assuming that the maximum number of 10+ minutes calls occurs, evaluate whether customer service staffing is adequate to handle all the calls without making customers wait on hold.
- Use the Poisson formula to evaluate whether it is financially viable to keep a store open 24 hours a day. Calculate the average number of sales made by the store during the extra overnight shift – the period from midnight to 8 A.M. Using the distribution formula then, calculate the probable lowest number of sales that might be made during the overnight shift.

Finally, determine whether that lowest probable sales figure represents sufficient revenue to cover all the costs (wages and salaries, electricity, etc.) of keeping the store open during that time period, plus provide a reasonable profit.

- Review and evaluate business insurance coverage. Determine the average number of losses or claims that occur each year and that are covered by the company’s business insurance. Then do a Poisson probability calculation to determine the maximum and minimum numbers of claims that might reasonably be filed during any one year.

Review the cost of your insurance and the coverage it provides. Consider whether perhaps you’re overpaying – that is, paying for a coverage level that you probably don’t need, given the probable maximum number of claims.

Alternatively, you may find that you’re underinsured – that if what the Poisson distribution shows as the probable highest number of claims actually occurred one year, your insurance coverage would be inadequate to cover the losses.

### Summary

The Poisson Distribution can be a helpful statistical tool you can use to evaluate and improve business operations. Excel offers a Poisson function that will handle all the probability calculations for you – just plug the figures in.

Learn more in CFI’s Financial Math Course.

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