## What is Poisson Distribution?

In the statistics, Poisson distribution refers to the distribution function which is used in analyzing the variance which arises against the occurrence of the particular event on an average under each of the time frames i.e., using this one can find the probability of one event in specific event time and variance against an average number of the occurrences.

Poisson Distribution Equation is given below:

**P (x; u) = (e**

^{-u}) * (u^{x}) / x!Where

- u = average number of occurrences during the time period
- P (x; u) = probability of x number of instances during the time period
- X= number of occurrences for which probability needs to be known

### Explanation

The formula is as follows-

P(x;u) = (e^-u).(u^x)/x!

Where

- u = average number of occurrences during the time period
- X= number of occurrences for which probability needs to be known
- P(x;u) = probability of x number of instances during the time period given u is an average number of occurrence
- e = Euler’s number which is the base of the natural logarithm, approx. value of e is 2.72
- x! = It is known as x factorial. Factorial of a number is a product of that integer and all integer below. For eg. 4! = 4*3*2*1

### Examples

#### Example #1

**Let us take a simple example of a Poisson distribution formula. The average occurrence of an event in a given time frame is 10. What would be the probability of that event occurrence for 15 times?**

In this example, u = average number of occurrences of event = 10

And x = 15

Therefore, the calculation can be done as follows,

P (15;10) = e^(-10)*10^15/15!

P (15;10) = 0.0347 = 3.47%

Hence, there is 3.47% probability of that event to occur for 15 times.

#### Example #2

Usage of the Poisson distribution equation can be visibly seen for improving productivity and operating efficiency of a firm. It can be used to find out whether it is financially viable to open a store 24 hours a day.

Let’s say Walmart in the U.S. is planning to open its store 24 hours a day. For finding out the viability of this option, at first Walmart management will find out the average number of sales between 12 midnight and 8 am. Now it will calculate its total operating cost for the working shift from 12 am to 8 pm. Based on this operating cost Walmart management knows that what is the minimum number of sales units to breakeven. Then with the Poisson distribution formula, it will find out the probability of that sales number and see whether it is viable to open the store 24 hours a day or not.

For Example, let’s say the average cost of operating on a day is $10,000 from 12 am to 8 pm. Average sales would be $10,200 at that time. For breakeven, each day sales should be $10,000. Now we will find out the probability of $10,000 or lower sales on a day so that breakeven can be achieved

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Therefore, the calculation can be done as follows,

P(10,000,10200) = POISSON.DIST(10200,10000,TRUE)

P(10,000,10200) =97.7%

Hence there is a 97.7% probability for $10,000 or lesser sell on a day. In the same way, there is a 50.3% probability for $10,200 or lesser dell on a day. That means between 10,000 and 10,200 sales probability is 47.4%. Hence there is a good chance for the firm to break even.

#### Example #3

Another use of the Poisson distribution formula is in Insurance Industry. A company which is in insurance business determines its premium amount based on the number of claims and amount claimed per year. So, to evaluate its premium amount, the insurance company will determine the average number of a claimed amount per year. Then based on that average, it will also determine the minimum and the maximum number of claims that can reasonably be filed in the year. Based on the maximum number of the claim amount and the cost and profit from the premium, the insurance firm will determine what kind if the premium amount will be good to break even its business.

Let’s say the average number of claims handled by an insurance company per day is 5. It will find out what is the probability of 10 claims per day.

Therefore, the calculation of the Poisson distribution can be done as follows,

P(10;5) = e^(-5). 5^10/10!

P(10;5) = 1.81%

Hence there is very little probability that the company will have to 10 claims per day and it can make its premium based on this data.

### Relevance and Uses

The Poisson distribution equation is very useful in finding out a number of events with a given time frame and known rate. Below are some of the uses of the formula:

- In the call center industry to find out the probability of calls which will take more than usual time and based on that finding out the average waiting time for customers.
- To find out the maximum and a minimum number of sales in odd hours and find out whether it is viable to open a store at that time.
- To find out the probability of a number of road accidents in a time interval.
- To find out the probability of the maximum number of patients arriving at a time frame,
- A number of maximum and minimum and clicks on a website.
- To find out visitors’ footfalls in a mall, restaurant, etc.
- To find out the probability of maximum and a minimum number of an insurance claim in a year.

### Poisson Distribution in Excel

It is very easy to find out Poisson distribution using excel. There is an excel function to find out the probability of an event. Below is the syntax of the function-

Where

- x= number of occurrences for which probability needs to be known
- Mean = average number of occurrences during the time period
- Cumulative= its value will be False if we need the exact occurrence of an event and True if a number of random events will be between 0 and that event.

We will take the same example 1 that we have taken above. Here x = 15, mean = 10 and we will have to find the probability of an exact number of events. So, the third argument will be false.

Hence P(15;10) = POISSON.DIST (15,10,FALSE) =0.0347 =3.47%

Here we got exact value using basic excel formula.

Let’s assume in the above example we need to find out the probability of occurrence between 0 to 15, then in the formula instead of false, we will use TRUE.

P(x<=15) = POISSON.DIST(15,10, TRUE) = 95.1%

That means the probability of occurrence of the event between 0 and 15 with 15 inclusive is 95.1%.

### Recommended Articles

This has been a guide to Poisson Distribution. Here we discuss how to calculate the Probability of X using Poisson distribution formula in excel with examples and downloadable excel template. You can learn more about financial modeling from the following articles –

- Poisson Distribution in Excel
- Formula of T Distribution
- Binomial Distribution Formula
- Sampling Distribution Formula
- APR vs APY

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