Poisson Distribution Calculator
Calculate Poisson probabilities for rare events. Find P(X = k), P(X ≤ k), and P(X > k) given an average rate λ.
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How to use this calculator
λ is the average number of events in the interval. k is the specific count of interest. e is Euler's number (~2.71828).
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Enter λ (lambda): the average number of events per interval (e.g. 3 calls per minute).
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Enter k: the specific count you want the probability for (e.g. exactly 4 calls).
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The calculator returns P(X = k), the cumulative P(X ≤ k), and P(X > k).
Frequently asked questions
What is the Poisson distribution used for?
The Poisson distribution models the number of independent events that occur in a fixed interval of time or space, when the average rate is known. Examples: call centre arrivals per minute, defects per product, accidents per year.
When does the Poisson distribution apply?
Use it when: events are independent; the rate is constant; two events cannot occur at exactly the same instant; and you're counting discrete events in a fixed interval.
How is Poisson different from binomial?
The binomial distribution models a fixed number of trials with probability p. The Poisson distribution models an unlimited number of possibilities with a small individual probability — it is the limiting case of the binomial as n → ∞ and p → 0 with λ = np fixed.
Poisson Distribution Calculator — P(X=k) & Cumulative Probability
How to use the poisson distribution
This poisson distribution gives you instant, accurate results — no registration or download required. Enter your values above and get your result in seconds. The tool is free, works on all devices, and keeps your data private — nothing is stored or shared.
How the poisson distribution works
The poisson distribution calculator uses standard formulas used in statistical analysis, data science, and research. Enter your inputs, and the tool calculates the result instantly in your browser. No server-side processing means your data stays on your device. Results update in real time as you change inputs.
Understanding the Poisson distribution
Named after Siméon Denis Poisson, this discrete probability distribution predicts the number of events in a fixed period when events happen at a constant average rate and independently of each other. Classic applications include queuing theory, insurance modelling, nuclear decay, and epidemiology.
Mean and variance of Poisson
A distinctive property of the Poisson distribution is that its mean and variance are both equal to λ. This makes it easy to estimate λ from data: just compute the sample mean. If the sample variance is much larger than the mean (overdispersion), a negative binomial distribution may be more appropriate.
Poisson distribution: how it works
Statistical analysis underpins data science, research, quality control, and business intelligence. This tool applies established textbook formulas to your dataset and returns results instantly, eliminating manual calculation errors.
Who uses this tool?
Researchers, students, analysts, and business professionals use it to understand datasets quickly. Whether you are summarising survey responses or checking experiment results, this tool delivers the key figures instantly.
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