Poisson Distribution Calculator

The Poisson distribution models the probability of a given number of independent events occurring in a fixed interval of time, space, or another continuum, given a known average rate λ. It applies when events are independent of one another, the average rate is constant throughout the interval, two events cannot occur at exactly the same instant, and the probability of an event in a tiny sub-interval is proportional to the sub-interval length. These conditions are met surprisingly often in real systems.

• ›Customer arrivals at a service counter (λ = 15 per hour)
• ›Radioactive decays per second from a known isotope sample
• ›Defects per metre of manufactured cable or textile
• ›Website page requests per second to a web server
• ›Number of mutations per generation in a genome replication
• ›Road accidents per month on a given stretch of highway

This is the Poisson PMF (probability mass function). λ is the average rate (events per interval), k is the exact count of interest, e ≈ 2.71828 is Euler's number, and k! is k factorial. The factor e⁻λ represents the probability of zero events occurring in a Poisson process with rate λ. The factor λᵏ/k! captures the combinatorial weight for exactly k independent events. The sum over all k from 0 to ∞ equals exactly 1, the probabilities form a proper distribution.

This is a defining and unique property of the Poisson distribution, provable from the moment generating function M(t) = exp(λ(eᵗ−1)). The mean E[X] = λ, and the variance Var(X) = λ, both equal the rate parameter. Physically, it arises because Poisson events are independent: each event contributes exactly 1 to both the expected count and the variance. If your data shows mean ≠ variance, the Poisson model is violated and a different distribution should be considered.

• ›Variance < Mean (underdispersion): events are more regularly spaced than random, e.g. scheduled arrivals
• ›Variance > Mean (overdispersion): events cluster together, e.g. accidents that attract more accidents
• ›Negative binomial: has Variance > Mean by design, handles overdispersed count data
• ›If Var/Mean ≈ 1 in your data, Poisson is likely a good fit; if Var/Mean >> 1, use negative binomial

Yes, λ is an average rate and can be any positive real number. Average rates rarely come out to round integers: λ = 2.5 calls per minute, λ = 0.37 accidents per month, or λ = 12.8 particles per second are all perfectly valid Poisson rate parameters. Only k (the number of events you are counting) must be a non-negative integer, because you cannot observe 2.5 events, only 0, 1, 2, 3, and so on. The Poisson PMF evaluates exactly for any positive real λ paired with any non-negative integer k.

• ›λ = 0.1: rare events, P(X=0) = 90.5%, P(X=1) = 9.0%, P(X≥2) = 0.5%
• ›λ = 1: moderate, P(X=0) = 36.8%, P(X=1) = 36.8%, equal probabilities for 0 and 1
• ›λ = 5: busier system, distribution peaks around k = 4–5, spreading noticeably
• ›λ = 20: large rate, distribution approximates a normal N(20, √20) bell curve

PMF (probability mass function) gives P(X = k), the probability of observing exactly k events. CDF (cumulative distribution function) gives P(X ≤ k), the probability of observing k or fewer events, computed as the sum of PMF values from 0 to k. The exceedance probability P(X ≥ k) = 1 − P(X ≤ k−1) = 1 − CDF(k−1). Choosing the right form depends on the question being asked.

• ›Use PMF for "exactly k" questions: P(exactly 3 defects in a batch)
• ›Use CDF for "at most k" questions: P(no more than 2 failures this week)
• ›Use exceedance for "at least k" questions: P(at least 5 customers arrive in the next hour)
• ›The bar chart shows the PMF; the cumulative percentage column in the table shows the CDF

If events occur as a Poisson process with rate λ (events per unit time), then the waiting time between consecutive events follows an Exponential distribution with mean 1/λ and rate parameter λ. These two distributions are duals of the same underlying memoryless process: Poisson counts how many events occur in a fixed time window; Exponential measures how long until the next event arrives. The memoryless property means the waiting time until the next event is independent of how long you have already waited.

• ›Poisson(λ = 5/hour) → inter-arrival times follow Exp(mean = 12 minutes)
• ›If calls arrive at λ = 10/min, P(wait > 30 sec) = e⁻⁵ ≈ 0.67%
• ›Both distributions are used together in queueing theory (M/M/1 queues)
• ›Gamma distribution: sum of k exponential waiting times (time until k-th event)

Overdispersion occurs when the observed variance in count data exceeds the mean, violating the core Poisson assumption that mean = variance. This is common in practice because many real-world count processes are heterogeneous: different observations have different underlying rates, or events cluster together rather than occurring independently. A Poisson model fitted to overdispersed data will underestimate the probability of very high and very low counts, producing poor predictions and overconfident confidence intervals.

• ›Common cause: hidden subgroups with different λ values (some customers are very frequent, others rarely arrive)
• ›Common cause: contagion effects (one traffic accident increases the risk of another)
• ›Detection: compute sample variance / sample mean; if >> 1, overdispersion is likely
• ›Fix: use negative binomial distribution, adds a variance parameter k, reducing to Poisson as k → ∞
• ›In R: glm(..., family=poisson) vs glm.nb(...), compare AIC to choose the better model