What is the Poisson distribution?

The Poisson distribution models the number of times a random event occurs in a fixed interval of time or space, given that events happen independently at a constant average rate λ (lambda). Classic examples include phone calls arriving at a switchboard per minute, radioactive decays per second, typographical errors per page, or bus arrivals per hour.

What is the Poisson PMF formula?

P(X = k) = (λ^k · e^(−λ)) / k!, where λ is the mean rate, k is the non-negative integer count of events, and e ≈ 2.71828. The calculator evaluates this in log-space (ln P = k·ln λ − λ − ln k!) for numerical stability across the full range λ = 0.001 to 200.

What does λ (lambda) represent?

λ is the average number of events in the interval you are studying. It is simultaneously the mean and the variance of the Poisson distribution — a unique property that makes the distribution easy to parameterise from real data. If your process runs over different time windows, scale λ proportionally: if 3 events happen per hour on average, then λ = 1.5 for a 30-minute window.

When does the Poisson distribution apply?

Use it when events are independent of each other, the average rate is constant over the interval, two events cannot occur at exactly the same instant (non-simultaneous), and you are counting occurrences (not measuring a continuous quantity). It is also the limiting case of the Binomial distribution when n is large, p is small, and λ = np stays finite.

How are the mean, variance and standard deviation related to λ?

For a Poisson random variable X with rate λ: Mean μ = λ, Variance σ² = λ, Standard deviation σ = √λ, Skewness = 1/√λ, Excess kurtosis = 1/λ. The equality of mean and variance is the defining signature of a Poisson process; if your data show σ² substantially larger than μ you likely have over-dispersion and a Negative Binomial model may fit better.

What is the mode of a Poisson distribution?

When λ is not an integer, the mode is floor(λ) — the largest integer not exceeding λ. When λ is a positive integer, both λ−1 and λ are modes (the distribution has two equally likely peaks). The calculator displays the lower of the two modes when λ is a whole number.

Poisson Distribution Calculator

The Poisson distribution is one of the most widely used probability models in statistics, engineering, biology, finance and operations research. It describes the number of independent random events that occur in a fixed interval when those events happen at a known, constant average rate. This calculator lets you compute any Poisson probability — exact, cumulative or range — instantly in your browser, alongside the full PMF/CDF table and a bar-chart view of the distribution.

How it works

The Poisson probability mass function (PMF) is:

P(X = k) = λ^k · e^(−λ) / k!

where λ (lambda) is the average number of events per interval, k is the non-negative integer count you want to evaluate, and e ≈ 2.71828 is Euler’s number. The calculator evaluates this in log-space:

ln P = k · ln λ − λ − ln(k!)

before exponentiating, which avoids floating-point underflow for large k or large λ — a common pitfall in naive implementations that compute λ^k and k! separately.

Cumulative queries use the CDF: P(X ≤ k) = Σᵢ₌₀ᵏ P(X = i). The “at least k” query uses the complement: P(X ≥ k) = 1 − P(X ≤ k−1). The range query combines both: P(k₁ ≤ X ≤ k₂) = P(X ≤ k₂) − P(X ≤ k₁ − 1).

Distribution moments for a Poisson(λ) random variable:

Statistic	Formula	Significance
Mean	λ	Expected number of events
Variance	λ	Equal to the mean — unique Poisson signature
Std deviation	√λ	Spread around the mean
Skewness	1/√λ	Always right-skewed; approaches 0 as λ → ∞
Excess kurtosis	1/λ	Heavier right tail than Normal for small λ

Worked example

A hospital emergency department receives, on average, 4.5 patients per hour during night shifts. The number of arrivals per hour follows a Poisson distribution with λ = 4.5.

Question: What is the probability that exactly 3 patients arrive in one hour?

Step 1 — Apply the PMF: P(X = 3) = (4.5³ × e^(−4.5)) / 3! = (91.125 × 0.01111) / 6 = 1.0125 / 6 ≈ 0.1687 (16.87%)

Step 2 — Cumulative check: P(X ≤ 3) = P(0) + P(1) + P(2) + P(3) ≈ 0.0111 + 0.0500 + 0.1125 + 0.1687 ≈ 0.3423 (34.23%)

Step 3 — At-least query: P(X ≥ 3) = 1 − P(X ≤ 2) ≈ 1 − 0.1736 ≈ 0.8264 (82.64%)

Query	Probability
P(X = 3)	16.87%
P(X ≤ 3)	34.23%
P(X ≥ 3)	82.64%
P(3 ≤ X ≤ 6)	61.48%

So the department should staff for at least 3 arrivals roughly 83% of the time.

Formula note

The Poisson distribution emerges as the limit of the Binomial(n, p) distribution as n → ∞ and p → 0 with np = λ held fixed. This approximation is accurate whenever n ≥ 100 and p ≤ 0.01. Conversely, when λ is large (λ ≥ 30), the Poisson distribution is well approximated by a Normal distribution with μ = σ² = λ — though the Poisson calculator here remains exact at any λ since it operates entirely in log-space.