Poisson Distribution Probability Calculator | PMF, CDF & Survival from λ and k
Enter the average rate λ and an observed count k to get the probability of exactly k events (PMF), at most k events (CDF), and at least k events (survival) in one view. Mean, variance, standard deviation, and mode appear alongside, so you can size a threshold or sanity-check an arrival rate without juggling formulas.
💡 About this tool
The Poisson distribution models how many times a random event happens in a fixed window of time or space when the average rate λ is known. Think calls per hour at a support desk, server failures per day, or defects per production batch — events that arrive independently at a roughly constant rate.
Doing this by hand means evaluating e^(-λ)·λ^k/k!, and the factorial overflows a pocket calculator once k gets large. This tool computes through the log-gamma function (a Lanczos approximation), so it stays numerically stable up to λ = 1000 and k = 2000 instead of returning Infinity or NaN. It also separates CDF from survival, which keeps you from confusing "exactly k", "at most k", and "at least k" — the three questions people mix up most.
If you run an SRE or SLO practice, the survival value P(X ≥ k) is the one you usually want: given an expected count λ, how likely is a burst that hits or exceeds k? The same setup covers queueing-theory arrival checks and basic quality-control defect rates.
🧐 Frequently Asked Questions
What is the difference between PMF and CDF? The PMF gives the probability of exactly k events, P(X = k). The CDF gives the cumulative probability of at most k events, P(X ≤ k). Survival, P(X ≥ k), covers k or more and is a different range than the CDF.
Why do the mean and variance both equal λ? That is a defining property of the Poisson distribution: E[X] = λ and V(X) = λ, with standard deviation √λ. Checking whether your sample mean and variance are close is a quick test of whether data is even Poisson-like.
How is the mode determined? When λ is not an integer, the mode is ⌊λ⌋, the floor of λ. It marks the single most likely count.
Can λ be a decimal? Yes. λ accepts decimals from 0.01 to 1000. The count k must be an integer from 0 to 2000.
How does Poisson relate to the binomial distribution? For a binomial with many trials n and small success probability p, the Poisson with λ = np is a close approximation — which is why Poisson is often called the law of rare events.
📚 Why Poisson shows up everywhere
The distribution is named after Siméon Denis Poisson, who published it in 1837 while studying the number of wrongful convictions in French courts. The pattern it captures is deceptively common: anything that happens independently at a steady average rate, counted over a fixed window, tends toward Poisson.
A useful gut check for engineers: because variance equals λ, the standard deviation is √λ, so the spread grows slower than the mean. Double your expected traffic and the relative noise actually shrinks — which is why high-volume systems feel more "predictable" per request even though absolute event counts swing more.