Poisson Distribution Calculator
The Poisson distribution counts how many independent events happen in a fixed interval when they occur at a constant average rate. Set the average rate λ (lambda), pick the question you want, and this tool returns the exact probability, a cumulative total, a range, or the inverse count, with a bar chart of the whole distribution and the region shaded. Everything runs in your browser.
P(X = 6) = 0.1042
How this is computed
ppois(), so deep tails and a large λ stay precise where a long running sum would drift.
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The five questions a Poisson answers
Each mode above maps to one R function. Exact counts use dpois(); cumulative totals use ppois(); the inverse uses qpois().
| You want | The event | In R |
|---|---|---|
| Exactly k events | P(X = k) | dpois(k, lambda) |
| At most k events | P(X ≤ k) | ppois(k, lambda) |
| At least k events | P(X ≥ k) | ppois(k - 1, lambda, lower.tail = FALSE) |
| Between a and b | P(a ≤ X ≤ b) | ppois(b, lambda) - ppois(a - 1, lambda) |
| The count for a probability | find k | qpois(prob, lambda) |
The formulas behind it
The probability of exactly k events is the Poisson mass function: the rate raised to the k, times the decay factor, divided by k factorial.
| Quantity | Formula | Meaning |
|---|---|---|
| P(X = k) | e-λ · λk / k! | Exact probability of k events |
| Mean | λ | Expected number of events |
| Variance | λ | Spread of the count (equal to the mean) |
| Std. dev. | √λ | Typical distance from the mean |
When λ ≥ 10 the count is close to a normal distribution with mean λ and variance λ, and a continuity correction (shifting the boundary by half a step) makes the approximation sharper. The Poisson also arises as the limit of a binomial with many trials and a small success probability, where n · p = λ; for a fixed number of trials use the binomial instead.
Frequently asked questions
What is the Poisson distribution used for?
It models how many independent events occur in a fixed interval of time or space when they happen at a constant average rate λ. Arrivals at a queue, calls to a help desk, defects per unit, or typos per page are classic cases. Set the average rate and the tool returns the probability of any count with R's dpois() and ppois().
What is the difference between P(X = k) and P(X ≤ k)?
P(X = k) is exactly k events, one bar of the distribution (dpois). P(X ≤ k) is k or fewer, the running total of every bar up to and including k (ppois). Use "Exactly k" for a single count and the cumulative modes for a threshold such as at most or at least.
When can I use the normal approximation to the Poisson?
A common rule of thumb is λ ≥ 10. When the mean is large, the Poisson is roughly Normal(λ, λ), and a continuity correction sharpens it. For a small rate the distribution is skewed; use the exact Poisson, which this calculator always does. The note under each result tells you which case you are in.
What does the inverse Poisson give me?
The inverse mode works backwards from a cumulative probability to a count: the smallest number of events whose cumulative probability reaches your target. That is R's qpois(), used for staffing limits, control-chart bounds, and the critical value of a Poisson test.
Does this calculator match R?
Yes. Every probability and quantile uses the same algorithms as R's dpois(), ppois() and qpois(), including the tail-accurate incomplete-gamma form, and agrees to at least nine decimals across the range, including k = 0, a rate below 1, and a large λ.