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.
Add this calculator to your site
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.
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.
P(X = x) = e-λλx x!
Where:
Suppose you work in a call center, where you receive an average of 4 calls per minute. Calculate the following probabilities:
Solution:
Given that:
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.
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 |
Poisson Distribution:
Binomial Distribution:
Poisson distribution is good for modeling independent events at a constant average rate within a specified interval.
Here are some general use cases:
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.
Support
Calculator Online Team Privacy Policy Terms of Service Content Disclaimer Advertise TestimonialsEmail us at
[email protected]© Copyrights 2024 by Calculator-Online.net