Poisson Distribution Calculator

Distributions and Statistical Models

Calculate the probability of a given number of events occurring in a fixed interval of time or space.

Examples

Explore real-world scenarios to understand how the Poisson distribution works.

Call Center Volume

call-center

A call center receives an average of 10 calls per hour. What is the probability of receiving exactly 5 calls in one hour?

λ: 10, x: 5

Manufacturing Defects

manufacturing

A factory produces light bulbs, and there are, on average, 2 defects per 100 bulbs. What's the probability of finding no defects in a batch of 100?

λ: 2, x: 0

Bacteria in a Sample

biology

A biologist expects to find 4 of a certain type of bacteria on a petri dish. What is the probability of finding at most 3 bacteria?

λ: 4, x: 3

Website Visitors

web-traffic

A website gets an average of 5.5 visitors per minute. What is the probability of getting more than 7 visitors in a minute?

λ: 5.5, x: 7

Other Titles
Understanding the Poisson Distribution: A Comprehensive Guide
Dive deep into the concepts, applications, and mathematics behind the Poisson Distribution Calculator.

What is the Poisson Distribution?

  • Core Concepts
  • Key Assumptions
  • Relationship to Other Distributions
The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. It's named after the French mathematician Siméon Denis Poisson.
Core Concepts
The distribution is defined by a single parameter, λ (lambda), which represents the mean number of events in the given interval. For example, if a call center averages 10 calls per hour, λ = 10.
Key Assumptions
For the Poisson distribution to be a valid model, several assumptions must hold: 1) Events are independent. 2) The average rate of events (λ) is constant. 3) Two events cannot occur at the exact same instant. 4) The probability of an event in a small interval is proportional to the length of the interval.
Relationship to Other Distributions
The Poisson distribution can be seen as a limiting case of the binomial distribution when the number of trials (n) is very large and the probability of success (p) is very small (i.e., n → ∞, p → 0, and np → λ).

Conceptual Examples

  • Number of emails you receive in an hour.
  • Number of typos on a page of a book.
  • Number of cars passing a specific point on a highway in a minute.

Step-by-Step Guide to Using the Poisson Distribution Calculator

  • Inputting Your Data
  • Interpreting the Results
  • Using the Reset and Example Functions
Our calculator simplifies the process of finding Poisson probabilities. Here's how to use it effectively.
Inputting Your Data
You need two pieces of information: the 'Average Rate of Success (λ)' and the 'Number of Successes (x)'. Lambda (λ) is the average number of times the event occurs, and x is the specific number you're interested in.
Interpreting the Results
The calculator provides several outputs: P(X = x) is the probability of exactly x events. P(X ≤ x) is the cumulative probability of x or fewer events. P(X ≥ x) is the probability of x or more events. It also shows the distribution's mean, variance, and standard deviation.
Using the Reset and Example Functions
The 'Reset' button clears all inputs and results. The 'Examples' section provides pre-filled scenarios to help you understand different use cases.

Real-World Applications of the Poisson Distribution

  • Finance and Insurance
  • Telecommunications
  • Quality Control
The Poisson distribution is not just a theoretical concept; it's widely used in various fields.
Finance and Insurance
Insurers use it to model the number of claims (e.g., car accidents, house fires) they expect to receive in a given period to set premiums appropriately.
Telecommunications
It helps in modeling the number of calls arriving at a call center or the number of data packets arriving at a router, which is crucial for capacity planning.
Quality Control
Manufacturers use the Poisson distribution to monitor the number of defects or flaws in a product (e.g., defects per square meter of fabric, blemishes per car panel).

Application Scenarios

  • Modeling the number of bankruptcies per month in a city.
  • Estimating the number of goals in a soccer match.
  • Predicting the number of radioactive decay events in a given time.

Mathematical Derivation and Formula

  • The Poisson Formula
  • Calculating Cumulative Probabilities
  • Mean, Variance, and Standard Deviation
The magic behind the calculator is the Poisson probability mass function (PMF).
The Poisson Formula
The probability of observing exactly x events is given by the formula: P(X=x) = (λ^x * e^-λ) / x! where 'e' is Euler's number (approximately 2.71828), 'λ' is the average rate, and 'x!' is the factorial of x.
Calculating Cumulative Probabilities
To find cumulative probabilities like P(X ≤ x), we sum the probabilities of all outcomes from 0 up to x: Σ [i=0 to x] P(X=i).
Mean, Variance, and Standard Deviation
A unique property of the Poisson distribution is that its mean (expected value) and variance are both equal to λ. The standard deviation is therefore √λ.

Calculation Example

  • If λ=3 and x=2, P(X=2) = (3^2 * e^-3) / 2! = (9 * 0.0498) / 2 ≈ 0.224.

Common Misconceptions and Correct Methods

  • Confusing with Binomial
  • Assuming Constant Rate
  • Misinterpreting Lambda
Understanding common pitfalls can help you apply the Poisson distribution correctly.
Confusing with Binomial Distribution
The Binomial distribution models the number of successes in a fixed number of trials, whereas the Poisson distribution models the number of events in a fixed interval. Use Binomial for 'out of n' scenarios and Poisson for 'rate' scenarios.
Assuming a Constant Rate
A key assumption is that the average rate λ is constant. If the rate changes over the interval (e.g., call volume is higher during business hours), a standard Poisson model may not be appropriate.
Misinterpreting Lambda
Ensure that the lambda (λ) you use corresponds to the interval you are interested in. If you know the rate per hour but want to calculate probability for a 30-minute interval, you must adjust λ accordingly (e.g., divide by 2).