Poisson Distribution Calculator
Calculate exact and cumulative Poisson probabilities
Enter the average rate of events (λ) and the number of successes (x) to compute all key Poisson probabilities instantly.
Poisson Distribution Calculator
Calculate exact and cumulative Poisson probabilities
About the Poisson Distribution Calculator
The Poisson distribution is one of the most important discrete probability distributions in statistics and applied mathematics. Named after French mathematician Siméon Denis Poisson, it describes the probability of a given number of events occurring in a fixed interval of time or space when those events happen independently and at a known constant average rate.
The distribution is completely characterized by a single parameter, λ (lambda), which represents the mean number of events in the given interval. For example, if a call center receives an average of 10 calls per hour, λ = 10. The probability of receiving exactly x calls in an hour follows the Poisson distribution with that lambda.
The Poisson probability mass function (PMF) is: P(X = x) = (e^−λ × λ^x) / x!, where e ≈ 2.71828 is Euler's number and x! is the factorial of x. This elegant formula allows us to compute exact probabilities for any non-negative integer x.
A remarkable property of the Poisson distribution is that its mean and variance are both equal to λ. This means that the standard deviation equals √λ. As λ increases, the distribution becomes more symmetric and approximates a normal distribution — a useful fact for large-scale applications.
This calculator computes five key probability values: P(X = x) for the exact count, P(X < x) for strictly fewer events, P(X ≤ x) for at most x events, P(X > x) for strictly more events, and P(X ≥ x) for at least x events. These cumulative forms are derived by summing the PMF over the relevant range.
The Poisson distribution is widely applied across science, engineering, finance, and medicine. Insurance companies use it to model the frequency of claims. Telecommunications engineers apply it to analyze call arrival rates and network packet flows. Quality control teams use it to model the number of defects per unit area. Epidemiologists employ it to model disease occurrence rates in populations.
The distribution also arises 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, with np = λ. This connection makes the Poisson model useful for rare-event modeling.
When using this calculator, ensure that the events you are modeling are truly independent and occur at a constant average rate. If the rate varies over your interval — for instance, web traffic is higher during business hours — a standard Poisson model may not be appropriate, and you may need a non-homogeneous Poisson process or a different distribution.
Examples
These examples demonstrate Poisson probability calculations for common real-world scenarios.
| Inputs (λ, x) | P(X = x) | Context |
|---|---|---|
| λ = 3, x = 2 | 0.22404 | Call center: avg 3 calls/min, P(exactly 2) |
| λ = 5, x = 4 | 0.17547 | Defects per unit: avg 5, P(exactly 4) |
| λ = 2, x = 0 | 0.13534 | Accidents per month: avg 2, P(zero accidents) |
| λ = 10, x = 8 | 0.11260 | Server requests: avg 10/sec, P(exactly 8) |
How to Use This Calculator
- Enter the average rate of events (λ) — this must be a non-negative decimal number, e.g., 3 or 2.5.
- Enter the number of events of interest (x) — this must be a non-negative whole number, e.g., 0, 1, 2.
- Click 'Calculate' to compute all five Poisson probabilities and the distribution statistics.
- Review P(X = x) for the exact probability and the cumulative values for range-based queries.
- Click 'Reset' to clear all fields and start a new calculation.
Frequently Asked Questions
What is the Poisson distribution?
The Poisson distribution is a discrete probability distribution that models the number of events occurring in a fixed interval of time or space. It is governed by a single parameter λ (lambda), the average number of events per interval. It applies when events are independent and occur at a constant mean rate.
What does λ (lambda) represent?
Lambda (λ) is the average number of events in the defined interval. For example, if a website receives an average of 50 visits per minute, λ = 50. Lambda must be a non-negative real number. Both the mean and variance of the Poisson distribution equal λ.
What is the difference between P(X = x) and P(X ≤ x)?
P(X = x) is the exact probability of observing precisely x events. P(X ≤ x) is the cumulative probability of observing x or fewer events, calculated by summing P(X = k) for k = 0 to x. Use the cumulative form when you need to know the chance of 'at most x' occurrences.
When should I use the Poisson distribution?
Use the Poisson distribution when you are counting the number of independent events in a fixed interval and the average rate is known and constant. Classic examples include call arrivals, radioactive decay counts, defect rates, and web server requests. If events are dependent or the rate varies, consider alternative models.
Can λ be a non-integer?
Yes — λ can be any non-negative real number, including decimals like 2.7 or 0.5. Only x (the number of successes) must be a non-negative integer. Fractional λ values arise naturally, for example when 3 events occur on average every 2 hours, giving λ = 1.5 per hour.
What is the relationship between the Poisson and Binomial distributions?
The Poisson distribution is a limiting case of the Binomial distribution. When the number of trials n is very large and the probability p of success per trial is very small, such that np → λ, the Binomial distribution converges to the Poisson distribution. This makes Poisson a useful approximation for rare-event counting in large populations.