Poisson Distribution Calculator
Counts per window when only the average rate λ is known: the point P(X = k) and both tails in log space, with the mean = variance signature and the overdispersion boundary stated.
Poisson Distribution Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- Independent arrivals at a steady rate — clustered counts (overdispersion) break the tail
- Direct log-space summation capped at λ = 100,000; beyond, approximations are named as approximations
In short: A help desk averages 2.5 tickets per hour: P(exactly 3) = e^−2.5·2.5³/3! = 0.213763, P(X ≤ 3) = 0.757576 and P(X ≥ 3) = 0.456187. The distribution’s signature is that its mean AND variance are both λ — here 2.5 and 2.5 — which is also how it fails: when your counts show a variance much larger than the mean, the events cluster (overdispersion) and the Poisson understates the tail. At λ = 4 the answer to “2 or fewer” is 0.238103. The arithmetic is log space through the Lanczos log-gamma, so λ = 10,000 prices P(X = 10,000) = 3.989390e-3 without a blink.
Formula
P(X = k) = e^−λ λ^k / k! · E[X] = Var(X) = λ
Computed as exp(k·lnλ − λ − lgamma(k+1)) — log space end to end, so neither λ^k nor k! ever gets formed as a plain number.
Worked Example
- Give the average rate λ (events per window) and the count k.
- Read the exact point P(X = k) and both tails.
- Check the signature card: Poisson means AND variances equal λ — if your observed counts disagree, the model is being tested.
- Rare-event limits: the binomial page hands off here when n is huge and p tiny with λ = np.
λ = 2.5, k = 3: P = 0.213763, P(≤ 3) = 0.757576, P(≥ 3) = 0.456187. λ = 4, k ≤ 2 → 0.238103. Large-λ spot check: P(10,000 events when λ = 10,000) ≈ 3.989390e-3.
Strengths & Limits Of This Model
Where this engine is strong
- Log-space end to end: pmf and tails exact from λ = 0.001 to 100,000
- The mean = variance signature and its failure mode printed on the page
Where it stops
- Single rate — no time-varying λ(t)
- No negative-binomial fallback for overdispersed counts (boundary stated)
Practical Use Cases
Ops
calls per hour, staffed against P(X > k)
Web
requests per second, burst tail
Epidemiology
cases per week vs a baseline rate
Methodology & Editorial Standards
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Poisson Distribution Calculator — 8 Expert FAQs
8 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
When is Poisson the right model?
When events arrive independently at a steady average rate and you are COUNTING per fixed window: calls per hour, typos per page, decays per second. If the rate itself drifts, or events cluster (one outage causes five alerts), the independence dies first — and the variance is what tattles.
Why do the mean and variance both equal λ?
That is the Poisson’s signature, not a coincidence: the whole distribution hangs on the one parameter, so location and spread are the same knob. Real counts with variance well above the mean are OVERDISPERSED — clustered arrivals — and the Poisson tail is then too short; the boundary card says so instead of pretending.
Where did the binomial go?
It collapsed into this: n → ∞ with p → 0 and np = λ fixed leaves exactly the Poisson. Rare events among many chances — typos in a book, wins in a lottery — are Poisson even when they are secretly binomial. The binomial page hands off here as n grows past what exact summation prices.
How big can λ be?
The arithmetic is direct log-space summation, honest to λ = 100,000 — pmf(10,000 at λ = 10,000) ≈ 3.989390e-3 costs nothing. Beyond the cap the page states the boundary rather than printing a normal tail as if it were exact.
What does P(X = 0) mean physically?
The quiet window: e^−λ. At λ = 2.5 that is 0.082085 — about one hour in twelve with no tickets at all. The zero-probability card is the natural sanity check for any claimed rate: if quiet hours never happen, λ is bigger than you think.
Can λ be a count, not a rate?
It is a rate per WINDOW: 2.5 per hour, 40 per kilometre, 9 per sheet. Double the window and you must double λ — the model’s scalability in time is the property that makes it the counting distribution. The window belongs next to every number you quote.
My counts burst — same answer?
No, and the signature catches it: clustering shows up as variance above the mean (overdispersion), and the Poisson tail — built on independence — understates exactly the big bursts you care about. The negative binomial owns that regime; this page names the boundary instead of faking the tail.
Why is this page paired with the exponential one?
Same process, two views: Poisson counts events per window; the exponential distribution times the GAPS between them, with the same λ. If tickets average 2.5 per hour, the wait until the next one is exponential at rate 2.5 — the exponential page prices the waiting question, including its memorylessness.