Poisson distribution: what it is and how to calculate it
Learn what it is and how to calculate the probability of a Poisson distribution.
Do you know what a statistical distribution is? What is it for or how to use it? Do you also know what a Poisson distribution is?
The Poisson distribution is nothing more than a statistical distribution. It was discovered by Siméon Denis Poisson and published together with his theory of probability in the year 1838.
To find the answers to the other questions expressed here and other curiosities, do not forget to read this post.
What is Poisson distribution?
The Poisson distribution is a discrete probability distribution applicable to occurrences of a number of events in a specific range.
To recognize a Poisson distribution, simply observe the following three aspects:
- The experiment calculates how many times an event occurs in a given time interval, area, volume, etc .;
- The probability of the event occurring is the same for each interval;
- The number of occurrences of one interval is independent of the other.
When to use the Poisson distribution?
When you start a Six Sigma project, the Green Belt or Black Belt, you should check what type of data (continuous or discrete) you are dealing with in the process output. This will determine which tools will be used in project development.
It is up to this professional to define which of the numerous statistical distributions is the one that best represents the process being studied. The statistical distributions can be divided into two large groups:
- Discrete Distribution (Attributes);
- Continuous Distribution (Variable).
Discrete distributions must be used to model situations where the output of interest can only assume integer values (discrete) such as number of heads or tails, 0 or 1 for failure or success, or 0,1,2, 3, ... as the number of occurrences of a particular event of interest for example.
The discrete distribution can still be divided into two families:
- Binomial Distribution;
- Poisson distribution.
Examples of Poisson distribution
In order to be clearer where and when to use this distribution, I have separated some examples, check it out!
- Computer users connected to the Internet;
- Clients coming to the cashier of a supermarket;
- Accidents with automobiles on a certain road;
- Typing errors for a certain period of time;
- Number of cars arriving at a gas station;
- Number of failures in components per unit of time;
- Number of requests for a server in a time interval.
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Now that you already know what a Poisson distribution is, what criteria to identify them, and also saw some examples, let's see how to calculate it.
How to calculate the Poisson distribution?
To calculate the probability of a Poisson distribution, you need the following probability function.
This equation is obtained through an approximation of the Binomial Distribution, considering 'n' tending to infinity and 'P' tending to zero.
Where x is the random variable that assumes the value k. μ is the value of the average rate and "e" is the Euler number, which is worth approximately 2.71828 ...
Poisson distribution: an applied example
In order to consolidate all this knowledge, I created an example to apply this concept, check it out!
In a bank, it was found after a data collection that the average number of customers who purchase certain insurance is 6 per hour. Determine, then, the probability that at a certain time of day exactly 8 insurances are sold.
In this context, it is possible to observe that the data have the three characteristics necessary for a Poisson distribution, so we will apply the data in the equation presented in this article and obtain the following expression:
That is, for a value of k = 8 and average μ = 6, we obtain a probability of 10% of a given hour being sold 8 insurance.
To further enhance our knowledge, I will expand upon this example. What would be the probability that at a given time less than 3 insurance would be sold?
It is important to realize that the probability of less than three in a discrete variable is equal to the probability of zero plus the probability of one plus the probability of two as shown below.
That is, the probability of selling less than three insurance at a given time is approximately 6.2%.
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