The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, assuming these events happen at a constant average rate λ and independently of each other. It's named after French mathematician Siméon Denis Poisson.

Lambda (λ) is the average rate of occurrence — the expected number of events in the given interval. For example, if a call center receives an average of 12 calls per hour, λ = 12. It is both the mean and the variance of the Poisson distribution.

Use the Poisson distribution when: (1) events occur independently, (2) the average rate is constant, (3) two events cannot occur at exactly the same instant, and (4) you're counting discrete occurrences in a continuous interval. Common applications include: call center volume, website traffic spikes, manufacturing defects, insurance claims, radioactive decay, and rare disease incidence.

The formula is: P(X = k) = (λ^k × e^−λ) / k! — where λ is the average rate, k is the number of occurrences, e ≈ 2.71828, and k! is the factorial of k. Our calculator uses a numerically stable recurrence method: P(X=0) = e^−λ, then P(X=k) = P(X=k−1) × λ / k, avoiding large factorial overflow.

The Binomial distribution models the number of successes in a fixed number of independent trials (n) with a constant success probability (p). The Poisson distribution models the number of events in a continuous interval with a known average rate (λ). The Poisson can be derived as a limiting case of the Binomial when n → ∞ and p → 0 while n×p = λ stays constant.

Mean = λ, Variance = λ, Standard Deviation = √λ. The mode is ⌊λ⌋ (the floor of λ) — or both λ and λ−1 when λ is an integer. As λ increases, the Poisson distribution approaches a normal distribution (by the Central Limit Theorem). For large λ (>30), a normal approximation with continuity correction is often used.

No. Lambda (λ) must be strictly positive. A λ of 0 would mean no events ever occur (degenerate distribution). Negative λ is mathematically undefined for the Poisson distribution. Our calculator enforces λ > 0 and will alert you if an invalid value is entered.

Our calculator uses double-precision floating-point arithmetic and a stable recurrence algorithm, providing excellent accuracy for λ ≤ 100 and k ≤ 200. For extremely large values (λ > 100), the recurrence may accumulate minor rounding errors, but results remain reliable to at least 6 decimal places. Beyond these bounds, consider using the normal approximation.