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Poisson Distribution Calculator

Enter the average rate of occurrence (λ), Poisson random variable (x), and select the type of probability (exact, cumulative, or complement) to find the probability of an event happening.

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This Poisson distribution calculator finds the probability of how often an event will likely occur within a fixed interval of time or space, given an average rate of occurrences(λ). It provides a step-by-step calculation and graph for a better understanding of discrete probability distributions.

What Is Poisson Distribution?

This distribution helps to predict the probability of how many times a specific number of events can occur within a fixed interval (space or time). 

Example: Imagine counting the number of people passing through a walkthrough gate in one minute. Poisson distribution helps determine the probability of a specific number of people passing through during the defined duration.

Properties of Poisson Distribution:

  • All the events occur independently of each other
  • Two events can not occur at the same time
  • Mean E(X) = Variance V(X) = λ
  • The average rate of occurrence (λ) remains constant over time, where np = λ
  • The value of the standard deviation is the same as the result of the square root of the mean

Poisson Distribution Formula:

P(X = x) = eλx x!

Where:

  • P(X = x) is the Probability of x occurrences
  • e indicates Euler's constant (approximately 2.71828)
  • λ (lambda) is the the average rate of occurrences
  • x shows the number of occurrences (poisson random variable)
  • x! is the factorial of x

How To Calculate Poisson Distribution?

  • Determine the average rate of occurrences
  • Write down the desired number of occurrences (x)
  • Calculate the factorial of x 
  • Put values in the Poisson distribution formula, solve the exponent part
  • After that divide the result by the factorial of x

Poisson Distribution (Solved Example):

Suppose you work in a call center, where you receive an average of 4 calls per minute. Calculate the following probabilities:

  • P(X = 3): Probability of receiving exactly 2 calls in a minute
  • P(X < 3): Probability of receiving less than 2 calls in a minute
  • P(X ≤ 3): Probability of receiving at most 2 calls in a minute

Solution:

Given that:

  • λ = 4 calls/minute

Probability P(x = 3):

Using the Poisson formula:

P(X = 3) = e-4*(4)3 3!

P(X = 3) = 0.018315 * 64 3 * 2 * 1

Poisson Distribution ≈ 0.19536

This means that the probability of having 3 calls is approximately 19.536 %

Calculating the probability P(x < 3) (For less than):

P(X = 0) = e-4*(4)0 0!

P(X = 0) ≈ 0.018315

P(X = 1) = e-4*(4)1 1!

P(X = 0) ≈ 0.07326

P(X = 2) = e-4*(4)2 2!

P(X = 2) ≈ 0.14652

P(X < 2) = P(X = 0) + P(X = 1) + P(X = 2) ≈ 0.018315 + 0.07326 + 0.14652 = 0.238095

The probability of having less than 3 calls per minute is approximately 0.238095 or 23.8095%. It indicates a low probability of having less than 3 calls per minute.

Calculate probability P(x ≤ 3) for each value of X:

 P(X = 0) ≈ 0.018315

P(X = 1) ≈ 0.07326

P(X = 2) ≈ 0.14652

P(X = 3) = e-4*(4)3 3!

P(X = 3) = 0.19536

P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)

P(X ≤ 3) ≈ 0.018315 + 0.07326 + 0.14652 + 0.19536 ≈ 0.433455

The probability of receiving less than or equal to 3 calls per minute is P(X≤ 3) ≈ 0.433455

Calculating Poisson probabilities manually can be time-consuming. To save time and simplify the calculation use our poisson distribution calculator. No matter, whether you are a beginner, student, researcher, or professional, the calculator can handle all your Poisson probability needs.

Poisson Distribution Table:

λ
X 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 0.9048 0.8187 0.7408 0.6703 0.6065 0.5488 0.4966 0.4493 0.4066 0.3679
1 0.0905 0.1637 0.2222 0.2681 0.3033 0.3293 0.3476 0.3595 0.3659 0.3679
2 0.0045 0.0164 0.0333 0.0536 0.0758 0.0988 0.1217 0.1438 0.1647 0.1839
3 0.0002 0.0011 0.0033 0.0072 0.0126 0.0198 0.0284 0.0383 0.0494 0.0613
4 0.0000 0.0001 0.0003 0.0007 0.0016 0.0030 0.0050 0.0077 0.0111 0.0153
5 0.0000 0.0000 0.0000 0.0001 0.0002 0.0004 0.0007 0.0012 0.0020 0.0031
6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0002 0.0003 0.0005
7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001

What Is The Difference Between Poisson Distribution And Binomial Distribution?

Poisson Distribution:

  • The variance is equal to the mean
  • The number of event occurrences is counted over a fixed time interval or space 
  • This is suitable for events happening independently at a constant rate

Binomial Distribution:

  • Each trial has Two possible outcomes (success or failure)
  • The number of times an experiment is repeated is known

When Do We Use Poisson Distribution?

Poisson distribution is good for modeling independent events at a constant average rate within a specified interval. 

Here are some general use cases:

  • Counting occurrences 
  • Rare events 
  • Quality control
  • Queueing systems

Easily calculate Poisson probabilities for these scenarios with the help of our Poisson distribution calculator. It can handle a variety of use cases, providing reliable results.

Reference:

From the source of Wikipedia: Probability mass function, Assumptions and validity

From the source of Investopedia: Understanding Poisson Distributions.

From the source of Brilliant ORG: Conditions for Poisson Distribution, Probabilities, Properties.

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