Poisson Distribution Calculator
Exact Poisson probabilities P(X = k), cumulative P(X ≤ k), P(X ≥ k), plus mean, variance and standard deviation — with a full distribution chart and step-by-step working.
The Poisson distribution explained, step by step
The Poisson distribution models the number of independent events that occur in a fixed interval when the average rate λ is constant. Each card below covers one piece of the formula this calculator uses.
1. λ is a rate — a count per interval, not a probability
λ (lambda) is the average number of events you expect over a fixed window of time, area or volume. It can be any positive real number — not bounded by 1 — and it scales with the window: doubling the interval doubles λ.
2. The PMF — probability of exactly k events
λ^k weighs scenarios producing k events, e^(−λ) normalises the distribution so probabilities sum to 1, and dividing by k! removes the double-count from arrival orders. The calculator uses log-gamma so k! stays stable even for large k.
3. Mean = variance = λ (signature property)
For any Poisson variable the mean and variance are both λ, so σ = √λ. If real data has variance much larger than its mean (over-dispersion) or much smaller, Poisson is the wrong model — that is the standard diagnostic.
4. When to switch models — binomial, exponential, normal
Fixed number of trials → binomial (Poisson is the limit as n → ∞, p → 0 with np = λ). Waiting time between events → exponential with rate λ — same process, different question. Once λ ≥ 20, Poisson(λ) is well approximated by Normal(λ, λ) with a continuity correction.
Features of this calculator
- Exact Poisson probabilities: P(X = k), P(X ≤ k), P(X < k), P(X ≥ k), P(X > k) — all in one calculation.
- Uses log-gamma internally, so probabilities stay accurate even when k! is astronomically large.
- Probability bar chart automatically spans a meaningful range of k values (roughly λ ± 4·√λ), with the bar for your chosen k highlighted.
- Mean, variance and standard deviation reported alongside the probabilities — with the useful reminder that mean = variance = λ.
- Show/hide step-by-step working with the formula, λ^k, e^(−λ) and k! all laid out.
Frequently asked questions
+What if I don't know λ?
Estimate it from data as the sample mean (total events ÷ number of intervals). If a call centre received 1,200 calls over 100 hours, an estimate of λ per hour is 12.
+How large can k be in practice?
Mathematically k has no upper limit. In practice P(X = k) becomes negligibly small once k is several standard deviations above λ. This calculator draws the chart out to about λ + 4·√λ, which covers essentially all the probability mass for reasonable λ.
+Can λ be 0?
Yes, but the distribution collapses: P(X = 0) = 1 and P(X = k) = 0 for any k ≥ 1. Not very useful in modeling terms.
+When can I use a normal approximation?
Once λ is large (a common threshold is λ ≥ 20), Poisson(λ) is well approximated by Normal(μ = λ, σ² = λ). Use a continuity correction if you approximate discrete probabilities from a continuous distribution.
+How is Poisson related to the exponential distribution?
If events happen according to a Poisson process with rate λ, the time between consecutive events follows an exponential distribution with rate λ. They're two views of the same underlying process.