Poisson Distribution , When it is used and how?

Poisson Distribution: It is used when we are given some old records and we are asked to generate forecast about any business or any sarvey. Big businessmen also use this concept to know when they need stock full and when at normal rate.
Practical Example of the Poisson Distribution
A store that rents books has an average rental of 200 books every Saturday night. Using this data, you can predict the probability that more books will sell (perhaps 300 or 400) on the following Saturday nights. Another example is the number of diners in a certain restaurant every day. If the average number of diners for seven days is 500, you can predict the probability of a certain day having more customers.

Because of this application, Poisson distributions are used by businessmen to make forecasts about the number of customers or sales on certain days or seasons of the year. In business, overstocking will sometimes mean losses if the goods are not sold. Likewise, having too few stocks would still mean a lost business opportunity because you were not able to maximize your sales due to a shortage of stock. By using this tool, businessmen are able to estimate the time when demand is unusually higher, so they can purchase more stock. Hotels and restaurants could prepare for an influx of customers, they could hire extra temporary workers in advance, purchase more supplies, or make contingency plans just in case they cannot accommodate their guests coming to the area.
With the Poisson distribution, companies can adjust supply to demand in order to keep their business earning good profit. In addition, waste of resources is prevented.

Calculating the Poisson Distribution
The Poisson Distribution pmf is: P(x; μ) = (e-μ * μx) / x!
The symbol “!” is a factorial.

μ (the expected number of occurrences) is sometimes written as λ. Sometimes called the event rate or rate parameter.
Sample question: The average number of major storms in your city is 2 per year. What is the probability that exactly 3 storms will hit your city next year?

Step 1: Figure out the components you need to put into the equation.
μ = 2 (average number of storms per year, historically)
x = 3 (the number of storms we think might hit next year)
e = 2.71828 (e is a constant)

Step 2: Plug the values from Step 1 into the Poisson distribution formula:
P(x; μ) = (e-μ) (μx) / x!
= (2.71828 – 2) (23) / 3!
= (0.13534) (8) / 6
= 0.180
The probability of 3 storms happening next year is 0.180, or 18%