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Poisson Distribution Calculator

Calculate the probability of a given number of events occurring in a fixed interval using the Poisson distribution. Models rare events like customer arrivals, defects, and radioactive decay.

Reviewed by Chase FloiedUpdated

This free online poisson distribution calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Average number of events per interval.

The specific number of events to find the probability for.

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Poisson Distribution Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Poisson Distribution Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

Formula Reference

Poisson Distribution Calculator Formula

See calculator inputs for the governing equation

Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.

When to Use This Calculator

  • •Use the Poisson Distribution Calculator when you need accurate results quickly without the risk of manual computation errors or unit conversion mistakes.
  • •Use it to verify calculations made by hand or in spreadsheets — an independent check can catch errors before they lead to costly decisions.
  • •Use it to explore how changing input parameters affects the output — a quick way to develop intuition and identify the most influential variables.
  • •Use it when collaborating with others to ensure everyone is working from the same numbers and applying the same assumptions.

About This Calculator

The Poisson Distribution Calculator is a free, browser-based calculation tool for engineers, students, and technical professionals. Calculate the probability of a given number of events occurring in a fixed interval using the Poisson distribution. Models rare events like customer arrivals, defects, and radioactive decay. It implements standard formulas and supports both metric (SI) and imperial unit systems with automatic unit conversion. All calculations are performed instantly in your browser with no data sent to a server. Use this calculator as a quick reference and sanity-check tool during design, analysis, and learning. Always verify results against primary engineering references and applicable standards for any safety-critical application.

About Poisson Distribution Calculator

The Poisson distribution calculator computes the probability of observing exactly k events in a fixed interval of time or space, given a known average rate lambda. Named after French mathematician Simeon Denis Poisson, this distribution is widely used to model rare or random events: the number of calls arriving at a call center per hour, the number of typos per page, the number of decay events per second from a radioactive source, or the number of car accidents at an intersection per month. The Poisson distribution assumes events occur independently and at a constant average rate. It is parameterized by a single value lambda, which is both the mean and the variance.

The Math Behind It

The Poisson distribution has the probability mass function P(X=k) = (lambda^k * e^(-lambda))/k! for k = 0, 1, 2, ... where lambda > 0 is the expected number of events. Its mean and variance are both equal to lambda, which provides a quick diagnostic: if the sample variance greatly exceeds the sample mean, the data may be overdispersed and a negative binomial distribution might be more appropriate. The Poisson distribution can be derived as the limit of the binomial distribution when n approaches infinity and p approaches zero such that np = lambda remains constant. The sum of independent Poisson random variables with parameters lambda_1 and lambda_2 is itself Poisson with parameter lambda_1 + lambda_2. The moment generating function is M(t) = exp(lambda*(e^t - 1)). In queueing theory, Poisson arrivals combined with exponential service times yield the M/M/1 queue, one of the most important models in operations research. The Poisson process, which generates Poisson-distributed counts, is the foundation of stochastic process theory and has applications from insurance claim modeling to genomics.

Formula Reference

Poisson PMF

P(X=k) = (lambda^k * e^(-lambda)) / k!

Variables: lambda = expected rate; k = number of events; e = Euler's number

Worked Examples

Example 1: Call center arrivals

A call center receives an average of 4 calls per minute. What is the probability of receiving exactly 6 calls in a given minute?

Step 1:lambda = 4, k = 6.
Step 2:P(X=6) = (4^6 * e^(-4)) / 6!.
Step 3:= (4096 * 0.01832) / 720 = 0.1042.

There is about a 10.4% probability of receiving exactly 6 calls.

Example 2: Website errors per day

A website averages 2 server errors per day. What is the probability of zero errors on a given day?

Step 1:lambda = 2, k = 0.
Step 2:P(X=0) = (2^0 * e^(-2)) / 0! = 1 * 0.1353 / 1 = 0.1353.

There is about a 13.5% chance of an error-free day.

Common Mistakes & Tips

  • !Using the Poisson distribution when events are not independent (e.g., cascading failures).
  • !Applying it when the rate is not constant over the interval (use non-homogeneous Poisson process instead).
  • !Forgetting that 0! = 1 when computing P(X=0).

Related Concepts

Binomial Distribution

The Poisson is the limiting case of the binomial for large n and small p.

Normal Distribution

For large lambda, the Poisson is well approximated by a normal distribution with mean and variance both equal to lambda.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Binomial Distribution Calculator

Frequently Asked Questions

What is the difference between Poisson and binomial distributions?

The binomial counts successes in a fixed number of trials with known probability. The Poisson counts events in a continuous interval with a known average rate. The Poisson is the limit of the binomial as n approaches infinity and p approaches zero.

Can lambda be a non-integer?

Yes. Lambda represents the average rate and can be any positive real number, such as 2.5 events per hour.

What if my variance is much larger than my mean?

This is called overdispersion and suggests the Poisson model is not a good fit. Consider the negative binomial distribution, which has an extra parameter for overdispersion.