How to Calculate Probability using the Poisson Distribution

During the Six Sigma Measure phase, Six Sigma DMAIC project Six Sigma Green Belt graduates work with two types data: continuous and discrete data. The Poisson distribution, a probability distribution for discrete information, takes the values X = 0, 1, 2, 3, and so forth. The Poisson distribution is used to describe data for which it is impossible to count all nonconformities. This is what Six Sigma online training participants will be able to see.
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The Poisson distribution describes defects data. These are non-conformities that affect a part of a product/service but do not render it unusable. Let’s examine how the Poisson distribution works.
The History
Simeon Denis Poisson, a French Mathematist-cum-Physicist, discovered the Poisson distribution in 1837. Poisson proposed the Poisson Distribution by using the example of modeling the number soldiers who were accidentally killed or injured by kicks from horses. The Poisson distribution was useful because it models uncommon events.
When is the Poisson Distribution appropriate?
The Poisson distribution is often used to model the number events (e.g., the number telephone calls at a business or intersection, the number at which an accident occurs, the number received by a call centre agent, etc.). In a given time period. The Poisson distribution can be used to determine the probability of occurrences per unit. This could be per-unit time, area, volume, etc. The Poisson distribution, or the probability distribution that results in a Poisson experiment, is simply this: The Poisson distribution can be used to analyze situations in which the number of trials and the probability that success are very low.
The Poisson Experiment
A Poisson experiment refers to a statistical experiment with the following properties:
The experiment produces results that can be classified either as successes or failures
It is known what the average number of success (m) in a given region is
The region’s size determines the likelihood of a successful outcome.
It is almost impossible to predict that a successful project will be realized in a very small area.

The Poisson parameter Lambda (l), which is the sum of all events (k) and the number of units (n), is called the equation: (l = (k/n).
The Formula
Below is the formula for Poisson distribution.

Let’s look at the components of the formula.
P(X =x) is the probability of x occurring in a given time interval
This symbol, ‘l’ or lambda, refers to the average number occurrences in the given interval
The number of desired occurrences is referred to as ‘x’.
The base of the natural algorithm is ‘e’. According to the Poisson distribution table the value of 0.0067 was calculated using the ‘l’ value and the ‘x’ values.

The Poisson distribution table can be downloaded online.
Illustration of Poisson distribution
The problem is related to the number of accidents at dangerous signals. To determine the effectiveness of safety measures at dangerous signals, it was decided that past records should be checked. These records show that there have been five accidents per week at this signal. We will calculate the probability of:
There are less than 2 accidents per semaine
More than 3 accidents per semaine

Calculating the probability of fewer that 2 accidents per week using Poisson distribution
We will need to answer the first question by using Poisson distribution to calculate the probability that there are fewer than two accidents per week. It can be expressed mathematically as P (X2). A probability of less than 2 means that there is a chance of zero accidents, and a possibility of one accident.

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