Poisson Distribution Formula

Introduction 

A Poisson experiment is a statistical experiment and a theoretical discrete probability that classifies the experiment into two categories, success or failure. Poisson distribution is a limiting process of the binomial distribution. The Poisson distribution formula is used to find the probability of events happening when we know how often the event has occurred.

What is Poisson Distribution Formula? 

Poisson Distribution Formula is used to show the number of times an event is likely to occur within a specified time duration. A Poisson random variable "x" is used to define the number of successes in the experiment. This distribution generally models the number of independent events within the given time interval. The Poisson distribution formula is very useful in situations where discrete events occur in a continuous manner.
The Poisson distribution is used under certain conditions:

• The number of trials, n, tends to infinity
• The probability of success, P, tends to zero
• np = 1 is finite

For a Poisson random variable, x = 0,1,2, 3,.........∞, the Poisson distribution formula is given by:

where,

• e is the Euler's number(e = 2.71828)
• x is a Poisson random variable that gives the number of occurrences(x= 0,1,2,.......)
• λ is an average rate of value in the desired time interval
• != factorial of functions

In shorthand notation, it is represented as X ~ P(λ)

The following properties of a Poisson model:

• The event or success is something that can be counted in whole numbers.
• The probability of having success in a time interval is independent of any of its previous occurrences.
• The average frequency of successes in a unit time interval is known.
• The probability of more than one success in a unit of time is very low.