Poisson Distribution Calculator

The Poisson distribution answers a single question very well: given an average rate of events, how likely is exactly k of them in a fixed window? It is the right model for things like calls per minute to a help desk, defects per kilometer of cable, server crashes per week, and goals per soccer match. Siméon Denis Poisson published the formula in 1837; Ladislaus Bortkiewicz famously validated it in 1898 by counting Prussian cavalry soldiers killed by horse kicks (the data fit beautifully, which made the distribution famous outside mathematics).

The formula:

P(X = k) = (λᵏ · e⁻λ) / k!

Where λ is the mean number of events per interval (the rate parameter), k is the count you want the probability for, and k! is k factorial. The distribution is defined for non-negative integers k = 0, 1, 2, …

Key properties:

• Mean = λ: the average count.
• Variance = λ: equal to the mean. This is one of the most distinctive Poisson features. If your real data has variance much larger than its mean, you have overdispersion, and a negative binomial fit is usually better.
• Mode: ⌊λ⌋ (floor of λ) when λ is non-integer; if λ is integer, both λ and λ-1 are modes.
• Standard deviation: √λ.
• Skewness: 1/√λ. Poisson is right-skewed for small λ, more symmetric for large λ.

For large λ (roughly λ > 20), Poisson is well-approximated by a Normal distribution with μ = λ and σ² = λ. For small λ (around 5 or below), the discrete bumps in the PMF are visible and the Normal approximation breaks down.

When Poisson works:

The classical assumptions are:

1. Events are independent. One event doesn’t make the next one more or less likely.
2. The rate λ is constant across the interval.
3. Two events cannot occur at exactly the same instant.
4. The probability of an event in a short interval is proportional to the length of the interval.

When these hold, the count of events follows Poisson exactly. When they don’t, you get under- or over-dispersion. Soccer goals are a good Poisson fit at the league level; goals in a single match where a team is desperately attacking late are not (the rate spikes near the final whistle).

Worked example, call center:

A call center receives an average of 4 calls per minute. What is the probability of exactly 6 calls in a given minute?

P(X = 6) = (4⁶ · e⁻⁴) / 6! = (4096 · 0.01832) / 720 = 75.03 / 720 = 0.1042 (about 10.4%)

What is the probability of at most 6 calls? Sum P(X = k) for k from 0 to 6:

P(X ≤ 6) ≈ 0.018 + 0.073 + 0.147 + 0.195 + 0.195 + 0.156 + 0.104 = 0.889 (about 88.9%)

So 88.9% of minutes will have 6 or fewer calls; 11.1% will have more.

Worked example, manufacturing defects:

A production line averages 1.5 defects per 1,000 units (λ = 1.5). What is the probability that a given 1,000-unit batch has zero defects?

P(X = 0) = (1.5⁰ · e⁻¹·⁵) / 0! = 1 · 0.2231 / 1 = 0.2231 (about 22.3%)

22% of batches will be defect-free. If your quality system promises “zero defects in 95% of batches,” you cannot deliver that with λ = 1.5; you’d need λ around 0.05.

Binomial vs Poisson:

When you have n independent trials each with success probability p, the count of successes follows Binomial(n, p). When n is large and p is small with np = λ moderate, Binomial(n, p) is well-approximated by Poisson(λ). Rule of thumb: use Poisson when n ≥ 20 and p ≤ 0.05. The approximation lets you skip computing big binomial coefficients.

This is also how Poisson “earned” its place in physics: counts of radioactive decays in a sample, photons hitting a detector, electrons emitted from a hot filament — these are all binomial in principle (n atoms each with a tiny decay probability), and Poisson in practice because n is enormous and p is tiny.

Confidence intervals for Poisson counts:

If you observed x events in an interval and want a 95% CI for the rate λ, the standard exact method uses the chi-squared distribution:

λ_low = 0.5 · χ²(0.025, 2x) λ_high = 0.5 · χ²(0.975, 2(x+1))

For x = 100 observed events, the 95% CI for the underlying rate is roughly 81 to 121. The square-root rule of thumb: λ ≈ x ± 2√x.

Compound and shifted Poisson:

Real-world processes often need extensions:

• Zero-inflated Poisson: many real datasets have more zeros than pure Poisson predicts (e.g., insurance claims, where most policies have zero claims). The zero-inflated model adds a separate probability mass at zero.
• Negative binomial: when variance > mean (overdispersion), this is the standard alternative. It has two parameters and handles real count data more flexibly than Poisson.
• Poisson regression: when the rate depends on covariates, you can model log(λ) as a linear function. Standard tool in epidemiology and quality control.

The single-parameter Poisson is rarely the final answer in serious modeling work, but it is almost always the first model you check.