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  • In statistics, Poisson Distribution is a discrete probability distribution that considers the number of successes per unit of continuous variable such as time/distance etc. over the course of many units.

  • Intervals that can be used are time, distance, area, volume etc

  • For a series of the discrete event, the average time between events is known, but the exact timing of the event is to occur is not know or is random.

  • The arrival of an event is independent of the previous event.

  • For example, the average number of visitors coming on a website in a day are 1000 but the number of visitors between 11 am, and 2 pm not known.

Probability mass function or of x events occurring in an interval – 

P(x) = (λx e-λ )/x!

e – Euler’s number – 2.718


E(X) = µ – Mean expected value

µ = λ = No. of time event occurs / Interval


Example – Number of visitors coming on the website in 1 hour or a number of Ford SUVs between New Jersey and New York.


Probability of occurrence is – 


P(x) = (λx e-λ )/x!

e – Euler’s number – 2.718

λ – Average Value

A cumulative mass function is the sum of all discrete probabilities – 

The probability of viewing fewer than 10 events in Poisson Distribution is : 

P(X, x <10) = ∑9i=0 (λx e-λ )/x!

(λ0 e-λ )/0! + (λ1 e-λ )/1!+ …. 


Then probabilities of seeing at least one is 


1 – the probability of occurrences none.


P(X, x >=1) = 1 – P(X:x=0)

1 – (λ0 e-λ )/0! = 1 – e-λ

  • In Poisson Distribution, the probability of success during the small time interval is proportional to the entire length of the interval.

  • If you know the expected value λ over an hour, then the expected value over 1 minute of that hour is – λminute = λ hour/60 

A quick exercise in understanding it better


A store sells on average 10 mobile phones in a week. 

  1. What is the probability of the store to sell 9 phones in the coming week?

  2. What is the probability of the store to sell less than 8 phones in the coming week?

  3. What is the probability of the store of not selling any phone on coming Tuesday?

August 16, 2019


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