**What is Poisson distribution?**

A Poisson distribution is generally defined as the tool to calculate the probability of the event when you know how often the event has occurred.

To explain in a simple manner, it gives the probability of the event generally happening in the fixed interval of time.**Attributes of Poisson distribution**

- The experiment results in the outcome that may be generally the success or failure.
- The average numbers of success that are supposed to occur in the specified region are known clearly.
- The probability of success is supposed to be proportional to the size of the region.
- The probability of success to occur within the extremely small region is virtually zero.

**The Formula for Poisson distribution**

Mathematically, the Poisson probability is given by,

P(X;µ) = (e^{-µ})(µ^{×})/x!

X = Generally the actual number of success that results from the experiment.

e = It is the constant approximately equal to 2.71828**Two Important Properties of Poisson Distribution**

The mean of the distribution is generally equal to µ.

The variance is also given by µ.**Example**

The average number of homes sold by the XXX realty company is 2 homes per day on an average. What is the probability that 3 homes will be sold by tomorrow?

P(*x*; μ) = (e^{-μ}) (μ^{x}) / x!

P (3;2) = (2.71828^{-2})(2^{3})/3!

= (0.13534)(8)/6

P (3;2) = 0.180**The Need of the Poisson distribution**

Poisson distribution is generally used to denote the probability for the rare events.**Example:**

- The number of defective pens per box of 5000 pens.
- The number of printing mistakes on each page while proofreading the book.

**Graphical Representation of the Poisson distribution**

All the Poisson distribution is generally skewed to right. This is the reason why Poisson distribution is called the probability of rare events.

**Advantages of Poisson distribution**

- It generally has an infinite number of trials.
- An unlimited number of outcomes is also possible.
- It can be generally used to test the independence.
- It can be used to predict the number of occurrences per unit, time and space.

**Fitting the Poisson distribution**

Generally, the process of fitting the Poisson distribution is very simple. We just need to obtain the m-value.

The list of other frequencies can be calculated by the below formula,

P_{o}= Ne^{-m}

P_{1}=Po.m/1

P_{2}=P_{1}.m/2

P_{3}=P_{2}.m/3 etc…**References**