Home›Calculators›Statistics›Poisson Distribution Calculator

Poisson Distribution Calculator

Statistics

Calculate Poisson probability P(X = k) for any event count in seconds. Enter the average rate (lambda) and number of events to get exact and cumulative probabilities.

01,000
01,000

P(X = k)

19.54%
P(X ≤ k)
43.35%
P(X ≥ k)
76.19%
Standard Deviation (σ)
2

This calculator computes your P(X = k), P(X ≤ k), P(X ≥ k), Standard Deviation (σ) from the values you enter.

Inputs
Average Rate (λ)Number of Events (k)
Outputs
P(X = k)P(X ≤ k)P(X ≥ k)Standard Deviation (σ)

What is a Poisson Distribution?

The Poisson Distribution Calculator computes the probability of a specific number of events occurring within a fixed interval, given a known average event rate (λ). Enter your average rate and the target number of events, and the calculator returns P(X = k), the cumulative P(X ≤ k) and P(X ≥ k), and the distribution's standard deviation.

The Poisson distribution is the standard tool for modeling rare, independent events occurring at a known average rate — customer arrivals, website errors, equipment failures, or accidents per time period. Unlike the binomial distribution, it requires only a single parameter (the average rate) rather than a fixed number of discrete trials.

For scenarios with a known, fixed number of trials and a per-trial success probability instead of a continuous rate, use the Binomial Distribution Calculator instead — the Poisson distribution is actually the limiting case of the binomial as trials become numerous and individual success probability becomes small.

How to use this Poisson Distribution calculator

  1. Enter the average rate (λ) — your historical average number of events per interval (per hour, per day, per batch), estimated from past data.

  2. Enter the target number of events (k) — the specific count you want to evaluate the probability of.

  3. Read P(X = k) — the probability of exactly that many events occurring.

  4. Check the cumulative probabilities — P(X ≤ k) for "at most" style capacity planning questions, or P(X ≥ k) for "at least" style risk questions.

  5. Compare to the standard deviation — √λ gives you a sense of how much natural variation to expect around your average rate.

  6. Re-estimate λ periodically — since the model assumes a constant rate, refresh your λ estimate periodically using recent historical data to keep predictions accurate as underlying conditions change.

Formula & Methodology

Poisson probability mass function:
P(X = k) = (e^−λ × λᵏ) / k!

Standard deviation:
σ = √λ

Variable definitions:
- λ (lambda) — the average rate of events per interval
- k — the target number of events being evaluated
- e — Euler's number (≈ 2.71828)

Worked example:

A customer support inbox receives an average of λ = 4 tickets per hour. What's the probability of receiving exactly 6 tickets in a given hour?

Step 1 — e^−4 ≈ 0.01832

Step 2 — 4⁶ = 4096, and 6! = 720

Step 3 — P(X = 6) = (0.01832 × 4096) / 720 ≈ 0.1042 (10.42%)

Step 4 — Standard deviation: σ = √4 = 2

A count of 6 tickets (2 above the mean of 4) is well within one standard deviation, so this outcome would not be considered unusual.

Assumption: This calculation assumes events occur independently at a constant average rate throughout the interval studied. If your actual event rate varies significantly within the interval (e.g., much higher during business hours), consider modeling separate intervals with their own estimated λ values instead of one combined average.

Frequently Asked Questions

A Poisson distribution models the probability of a given number of events occurring within a fixed interval of time or space, based on a known average rate (λ), assuming events occur independently and at a constant average rate. Classic examples include the number of customer arrivals at a store per hour, the number of typos per page, or the number of website server errors per day.
P(X = k) = (e^−λ × λᵏ) / k!, where λ is the average rate and k is the specific number of events you're evaluating. For example, if a call center averages λ = 4 calls per minute, the probability of getting exactly 6 calls in a given minute is P(X=6) = (e^−4 × 4⁶) / 6! = (0.0183 × 4096) / 720 ≈ 0.1042, or about 10.4%.
Lambda (λ) is the average rate of occurrence over your chosen interval — for example, average calls per minute, average defects per batch, or average accidents per month. It's typically estimated from historical data by dividing the total observed event count by the number of intervals observed (e.g., total calls received divided by total minutes observed).
Events must occur independently of each other, at a roughly constant average rate over the interval studied, and it must be theoretically possible for any number of events (however unlikely) to occur in the interval. Poisson modeling breaks down if events cluster together (violating independence) or if the rate genuinely changes significantly within the interval being studied (e.g., rush hour vs. midnight call volume).
For a Poisson distribution, the standard deviation is simply the square root of lambda: σ = √λ. This is a distinctive property of the Poisson distribution — unlike most distributions, its spread is completely determined by its mean, with no separate variance parameter to estimate.
The Poisson distribution is the limiting case of the [Binomial Distribution](/binomial-distribution-calculator/) as the number of trials (n) becomes very large and the probability of success (p) per trial becomes very small, while the product n×p (the average number of successes) stays constant and equal to λ. This is why Poisson is used for 'rare events over many opportunities' scenarios, like accidents per day across a large population, rather than fixed small-trial-count scenarios.
Unlike the binomial distribution, which caps the number of successes at the fixed number of trials (n), the Poisson distribution has no upper bound on k — because it doesn't model a fixed number of discrete trials, only a rate of occurrence over a continuous interval. In principle, any non-negative integer count of events (however improbable) is theoretically possible.
P(X ≤ k), the cumulative probability, tells you the probability of k or fewer events occurring — useful for questions like 'what's the probability of receiving no more than 3 support tickets today?' This is generally more actionable for capacity planning than the exact P(X = k), since real-world decisions usually depend on staying under (or exceeding) a threshold rather than hitting one specific count.
Call centers, emergency rooms, and IT support teams use Poisson probabilities to estimate the likelihood of receiving more requests than they can handle in a given period, given a known historical average arrival rate. This informs staffing decisions — for example, sizing a support team so that the probability of exceeding capacity in any given hour stays below an acceptable threshold like 5%.
Yes, and if it does, treating the process as a single constant-rate Poisson distribution becomes inaccurate. Many real processes have a rate that varies predictably (e.g., higher website traffic during business hours than overnight) — in these cases, analysts typically split the data into separate time segments, each with its own estimated λ, and apply the Poisson model separately within each segment.
A site reliability engineering team observing an average of 2 server errors per hour (λ = 2) could use this calculator to estimate the probability of experiencing 5 or more errors in a given hour (a potential incident threshold), helping set data-driven alerting thresholds rather than arbitrary ones — informing whether a spike of 5 errors is a genuine anomaly or within normal statistical variation.
This calculator computes probabilities using a numerically stable logarithmic gamma function approach rather than computing factorials directly, so it remains accurate even for large k values (hundreds) that would otherwise cause floating-point overflow in a naive factorial-based calculation.
Also known as
poisson probability calculatorpoisson distributionrare event probability calculatorlambda probability calculatorpoisson pmf calculator