Poisson Distribution Calculator

A Poisson distribution is defined as how many times an event occurs within x fixed interval. Interval can be minutes, hours, days or any unit of time. Not just time - it can be any interval - for example the number of misspellings in a page.

Let's take a practical example - Customer care executive gets three phone calls every 15 mins. What is the probability that he/she would get more than 5 calls in the next 30 minutes? Given the average rate of success is 3 calls for every 15 minutes. Therefore, average rate of success (λ) for our problem statement which is to calculate for 30 minutes would be 3 x (30/15) = 6. Number of occurences (x) is 5. It is also called Poisson random variable.


P(X < ) means probability of less than occurences. Similarly P(X > ) refers to probability of more than occurences.
Poisson Distribution Table

Probabilities of exactly N occurences from 0 through are shown below.

Important Points
  • Average rate of success (λ) must be greater than or equal to 0.
  • Poisson random variable must be a whole number, can't be in decimals.
  • Poisson random variable can't be less than zero.

Poisson Distribution has the following properties -

  • Average rate of success must be already known.
  • Probability of a single occurence is proportional to the size of the interval.
  • Probability of more than one occurence within a very short interval is small.
  • Operations team receives on average 113 complaints daily. What is the probability that the team would receive more than 137 complaints daily?
  • Colleges were closed 100 days every year in the first and second years of pandemic. What is the probability that colleges would be closed for 150 days next year (assuming next wave of pandemic would come next year and other factors constant)?
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