# Poisson distribution: how likely is k?

Computes Poisson probabilities P(X = k), P(X < k), P(X ≤ k), P(X > k), and P(X ≥ k) for a mean rate λ, with the mean, variance, and standard deviation.

- Page: https://www.acalculator.org/statistics/poisson-distribution-calculator
- JSON spec: https://www.acalculator.org/statistics/poisson-distribution-calculator.json
- Version: 50fe64ff6e82

## Default answer

Example with the default inputs (Average rate (λ) 3, Number of events (k) 5): With an average of 3, the chance of exactly 5 events is 0.100819, and of at most 5 is 0.916082.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| lambda | Average rate (λ) | The average number of events in the interval, more than 0 and up to 10,000. |
| k | Number of events (k) | The number of events to find the probability of, a whole number from 0 to 100,000. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| exactly | P(X = k) | The probability of exactly k events. |
| below | P(X < k) | The probability of fewer than k events. |
| atMost | P(X ≤ k) | The probability of k or fewer events. |
| above | P(X > k) | The probability of more than k events. |
| atLeast | P(X ≥ k) | The probability of k or more events. |
| mean | Mean (λ) | The expected number of events: λ. |
| variance | Variance (λ) | The variance of the number of events: λ. |
| sd | Standard deviation (√λ) | The square root of λ. |

## Method

P(X = k) = e^−λ × λᵏ ÷ k!; P(X ≤ k) = Σ P(X = i) for i = 0 … k; mean = variance = λ.

## Assumptions

- Events happen one at a time, independently, at a constant average rate λ over the interval.
- P(X ≤ k) and P(X ≥ k) come from the regularized incomplete gamma function, which equals the sum of the terms; each tail is computed directly, not as 1 minus the other, so tiny tails keep their precision.

## Worked examples

1. lambda = 3, k = 5 gives exactly = 0.100819, atMost = 0.916082, below = 0.815263, above = 0.083918, atLeast = 0.184737, sd = 1.732051. Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.19 Poisson Distribution. https://www.itl.nist.gov/div898/handbook/eda/section3/eda366j.htm.
2. lambda = 2, k = 0 gives exactly = 0.135335, below = 0, atLeast = 1, atMost = 0.135335. Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.19 Poisson Distribution. https://www.itl.nist.gov/div898/handbook/eda/section3/eda366j.htm.
3. lambda = 1, k = 2 gives exactly = 0.18394, atMost = 0.919699, above = 0.080301. Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.19 Poisson Distribution. https://www.itl.nist.gov/div898/handbook/eda/section3/eda366j.htm.
4. lambda = 0.5, k = 10 gives exactly = 0, atLeast = 0. Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.19 Poisson Distribution. https://www.itl.nist.gov/div898/handbook/eda/section3/eda366j.htm.
5. lambda = 100, k = 100 gives exactly = 0.039861, atMost = 0.526562. Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.19 Poisson Distribution. https://www.itl.nist.gov/div898/handbook/eda/section3/eda366j.htm.

## FAQ

### What is a Poisson distribution?

It gives the chance of each number of events in a fixed interval when events happen one at a time, independently, at a steady average rate λ. Examples are calls to a help desk per hour, typos per page, or cars through a toll gate per minute.

### What is the Poisson formula?

P(X = k) = e^−λ × λᵏ ÷ k!. With λ = 3 and k = 5, that is e^−3 × 243 ÷ 120 = 0.1008.

### What is the difference between P(X ≤ k) and P(X < k)?

P(X ≤ k) includes k itself; P(X < k) stops at k − 1. They differ by P(X = k). With λ = 3, P(X ≤ 5) = 0.9161 and P(X < 5) = 0.8153.

### What are the mean and variance of a Poisson distribution?

Both equal λ, so the standard deviation is √λ. If a shop gets 3 customers a minute on average, the count per minute has mean 3 and standard deviation 1.732.

### When can I use the Poisson distribution instead of the binomial?

When there are many trials, each with a small chance, the binomial with n trials and chance p is close to a Poisson with λ = n × p. A common rule of thumb is n of 20 or more and p of 0.05 or less.

### Can λ be a decimal?

Yes. λ is an average, so 2.5 events per hour is fine. k must be a whole number, because you count events.

### How precise are very small probabilities?

Each tail is computed directly from the incomplete gamma function, not as 1 minus a number close to 1, so a tail like P(X ≥ 10) with λ = 0.5 keeps its digits (1.7097 × 10⁻¹⁰).

## Sources

- NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.19 Poisson Distribution (probability mass function, cumulative distribution function, mean λ, standard deviation √λ), retrieved 2026-10-01. https://www.itl.nist.gov/div898/handbook/eda/section3/eda366j.htm