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Enter a rate λ and event count k to compute Poisson probabilities instantly.
What Is the Poisson Distribution?
The Poisson distribution is a discrete probability distribution that models the number of times a random event occurs within a fixed interval of time, space, or volume — when those events happen independently and at a constant average rate. Named after French mathematician Siméon Denis Poisson, it is one of the most widely applied distributions in statistics, engineering, and natural science.
It answers questions of the form: "If calls arrive at a rate of 3 per minute, what is the probability that exactly 5 calls arrive in the next minute?" The single parameter λ (lambda) defines both the average event rate and, uniquely, the full shape of the distribution. Because the mean and variance are both equal to λ, the Poisson distribution is easy to identify and parameterise from real data.
The Poisson Distribution Formula
The probability mass function (PMF) gives the probability of observing exactly k events:
Poisson Formula Components
Controls how sharply probability falls as λ shrinks. For λ = 3: e⁻³ ≈ 0.0498.
Raises the rate to the power of k. Grows with k until the factorial in the denominator dominates.
Divides out the number of arrangements so probabilities across all k sum to exactly 1.
Average number of events per interval. Also equals the mean and the variance of the distribution.
The specific count you are evaluating. Must be a non-negative integer: 0, 1, 2, 3, …
Euler's number — the base of the natural logarithm. Appears because the Poisson process involves exponential decay.
How to Calculate Poisson Probability
To compute P(X = k), multiply e raised to the negative rate (e−λ) by the rate raised to the target count (λk), then divide by k factorial (k!). In practice, use logarithms to avoid overflow: compute log P = −λ + k·ln(λ) − ln(k!), then exponentiate. This keeps computations numerically stable even for large values of k.
Example — a call centre handles 3 calls per minute on average (λ = 3). What is the probability of exactly 2 calls in the next minute? P(X = 2) = e⁻³ × 3² ÷ 2! = 0.0498 × 9 ÷ 2 ≈ 0.2240, or 22.40%. The cumulative probability P(X ≤ 2) sums P(0) + P(1) + P(2) ≈ 0.0498 + 0.1494 + 0.2240 ≈ 42.32%.
P(X = k), P(X ≤ k), and P(X ≥ k) Explained
P(X = k) is the exact probability — the probability that precisely k events occur and no more, no fewer. This is what the PMF formula computes directly.
P(X ≤ k) is the cumulative distribution function (CDF) — the probability that the count is k or any smaller value. It is computed by summing the PMF from 0 up to k. This answers "at most k" questions, such as "what is the chance of no more than 4 defects per unit?"
P(X ≥ k) is the survival function — the probability that the count is k or any larger value. It equals 1 − P(X ≤ k − 1). This answers "at least k" questions, such as "what is the chance of 5 or more server errors per hour?"
The remaining two variants are strict inequalities: P(X < k) = P(X ≤ k − 1), and P(X > k) = 1 − P(X ≤ k). All five values are displayed in the calculator results.
Common Poisson Distribution Examples
Call centres: If a helpdesk receives an average of 6 calls per hour, the Poisson distribution gives the probability of exactly 4, 8, or any other count arriving in the next hour. Staffing models use these probabilities to set the minimum number of agents needed to meet a service level target.
Server and network reliability: An application logging 0.5 critical errors per day follows a Poisson model. P(X = 0) = e⁻⁰·⁵ ≈ 60.65% — roughly a 60% chance of a clean day. Operations teams use this to set alert thresholds and estimate mean time between failures.
Manufacturing quality control: If a production line produces an average of 2 defective items per batch, P(X ≤ 3) tells the inspector the probability of passing a batch with at most 3 defects. Acceptance sampling plans are built directly on the Poisson CDF when lot sizes are large.
Epidemiology: Rare disease incidence expressed as cases per 100,000 people per year is well-modelled by the Poisson distribution when the underlying exposure is roughly uniform across the population. Public health researchers use this to identify clusters that exceed the expected background rate.
Ecology: The number of organisms per unit area in a randomly sampled field often follows a Poisson distribution when individuals settle independently. Biologists use goodness-of-fit tests against the Poisson PMF to detect clustering or regularity in spatial patterns.
Poisson Distribution Properties
Three properties make the Poisson distribution distinctive. First, the mean and the variance are both equal to λ — this is the unique equidispersion property. When analysed data shows variance significantly greater than the mean (overdispersion), a negative binomial model is usually more appropriate.
Second, the sum of independent Poisson random variables is also Poisson. If X₁ ~ Poisson(λ₁) and X₂ ~ Poisson(λ₂), then X₁ + X₂ ~ Poisson(λ₁ + λ₂). This additivity makes Poisson models easy to combine across time intervals or geographic regions.
Third, for large λ, the Poisson distribution converges to a normal distribution with mean λ and standard deviation √λ. The normal approximation is typically acceptable when λ ≥ 10, allowing the well-known z-score and confidence interval methods to be applied.
Poisson Distribution Reference Table
| k | λ = 1 | λ = 2 | λ = 3 | λ = 5 |
|---|---|---|---|---|
| 0 | 36.79% | 13.53% | 4.98% | 0.67% |
| 1 | 36.79% | 27.07% | 14.94% | 3.37% |
| 2 | 18.39% | 27.07% | 22.40% | 8.42% |
| 3 | 6.13% | 18.04% | 22.40% | 14.04% |
| 4 | 1.53% | 9.02% | 16.80% | 17.55% |
| 5 | 0.31% | 3.61% | 10.08% | 17.55% |
Poisson Distribution FAQ
What is the Poisson distribution used for? +
The Poisson distribution models counts of rare or independent events over a fixed interval — calls per minute, errors per hour, defects per unit, or species per quadrant. Whenever events occur randomly at a constant rate and cannot cluster by design, the Poisson model is the natural starting point for probability calculations and inference.
What is λ (lambda) in the Poisson distribution? +
Lambda is the single parameter of the Poisson distribution. It represents the average number of events expected in the chosen interval. Because the Poisson distribution has the equidispersion property, λ equals both the mean and the variance. To estimate λ from data, compute the sample mean of your count observations.
What is the difference between P(X=k), P(X≤k), and P(X≥k)? +
P(X=k) is the exact probability of precisely k events — computed directly from the PMF formula. P(X≤k) is the cumulative probability of k events or fewer — the sum of the PMF from 0 to k. P(X≥k) is the survival probability of k events or more — equal to 1 minus P(X≤k−1). Each answers a different practical question: exact, at-most, and at-least.
Why is the mean equal to the variance in a Poisson distribution? +
This follows mathematically from the Poisson process assumptions: events occur independently, the rate is constant, and simultaneous events have zero probability. In derived form, both the expected value and the variance of a Poisson random variable simplify to λ. If your data shows variance much larger than the mean (overdispersion), a negative binomial distribution usually fits better.
When can I use the Poisson distribution instead of the binomial? +
The Poisson distribution approximates the binomial well when the number of trials n is large and the success probability p is small, so that the mean np = λ stays moderate. A common guideline is n ≥ 20 and p ≤ 0.05. The Poisson is preferred in those settings because it depends on only one parameter (λ) rather than two (n and p), and it extends naturally to continuous-time processes.
What happens when λ is very large? +
As λ increases, the Poisson distribution becomes increasingly symmetric and bell-shaped. For λ ≥ 10, a normal distribution with mean λ and standard deviation √λ provides a good approximation, allowing standard z-score calculations. This calculator supports λ up to 1,000; above that, the normal approximation is more practical.
Can k be zero in the Poisson distribution? +
Yes — k = 0 is a valid and often meaningful event count. P(X = 0) = e^(−λ) gives the probability that no events occur in the interval. For λ = 0.5, P(X = 0) ≈ 60.65%, meaning there is a better-than-even chance of a quiet interval. Setting k = 0 in the calculator returns this probability directly along with all five cumulative and survival variants.
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