Poisson Probability Distribution Calculator (PMF, CDF & Rate Modeling

Statistical Foundations of the Poisson Distribution

Named after French mathematician Siméon Denis Poisson (1837), the Poisson distribution models the number of times a rare, independent event occurs within a specified fixed interval of time, area, distance, or volume:

• The Three Poisson Assumptions:

• Independence: The occurrence of an event in one interval does not influence occurrences in disjoint intervals.

• Constant Rate: The average rate of events (λ) remains constant per unit time/space.

• Simultaneity Impossibility: The probability of two or more events occurring at the exact same infinitesimal instant is zero.

• The Equidispersion Signature: A defining mathematical hallmark of a true Poisson process is that the mean is identically equal to the variance). When σ^2 > μ (overdispersion), the Negative Binomial distribution is used instead.

• Poisson as the Limit of Binomial (Law of Rare Events): When n ∞ and p 0 such that \ remains constant, the Binomial distribution B(n, p) converges mathematically to the Poisson distribution Pois(λ).

How to Use This Calculator

Enter your Average Event Rate (λ - Lambda / Expected Occurrences per Interval) (e.g. 4.5 (Arrivals per hour, defects per unit, calls per minute)), Target Number of Occurrences (k / x) (e.g. 3 (Exact event count to evaluate)), and Probability Event Type to Calculate — choose from Exact Probability: P(X = k), Cumulative Probability: P(X ≤ k) [At Most k], Upper Tail Probability: P(X ≥ k) [At Least k]. Results update instantly as you type — adjust any value to compare scenarios.

The Formula

Poisson Probability Distribution X Pois(λ) equations:

Probability Mass Function (PMF): P(X = k) = λ^k e^-λk!

Cumulative Distribution (CDF): P(X k) = Σ_i=0^k λ^i e^-λi!

Upper Tail Probability: P(X k) = 1 - P(X k - 1) = Σ_i=k^∞ λ^i e^-λi!

Mean \& Variance Equality (Equidispersion): E[X] = μ = λ, Var(X) = σ^2 = λ

Standard Deviation: σ = √(λ), Skewness: γ_1 = (1) / (√(λ))

Worked Example

For example, evaluating an emergency call center that receives an average rate of \ calls per minute: The probability of receiving exactly \ calls in a given minute is \ = 4.5^3 e^-4.53! = (91.125 × 0.011109) / (6) = 0.1687 (16.87\%). The probability of receiving at most 3 calls is P(X 3) = P(0) + P(1) + P(2) + P(3) = 0.0111 + 0.0500 + 0.1125 + 0.1687 = 0.3423 (34.23\%). Mean and variance are both 4.5, with .

Understanding Your Result

Statistical results require context: a mean alone is misleading without knowing the standard deviation (spread). Normal distribution reference: 68% of values fall within ±1 SD of the mean, 95% within ±2 SD, 99.7% within ±3 SD. For small sample sizes (n < 30), use t-distributions rather than z-scores. Statistical significance (p < 0.05) does not imply practical significance — effect size (Cohen's d, R², odds ratio) measures real-world importance. Correlation coefficient r: |r| < 0.3 weak, 0.3–0.7 moderate, > 0.7 strong.