Calculate Poisson distribution probabilities, cumulative distribution, expected value, variance, and visualize the PMF. Ideal for rare event modeling.
The Poisson Distribution Calculator computes exact and cumulative probabilities for Poisson-distributed random variables. Enter the average rate (λ) and the number of events (k) to find P(X = k), P(X ≤ k), P(X ≥ k), and view the complete probability mass function. It is especially useful when you need to compare one observed count against a known average without building a custom probability model first. That makes it a quick fit for planning, diagnostics, and sanity-checking observed counts.
The Poisson distribution models the number of events in a fixed interval when events occur independently at a constant average rate. Classic applications: calls per hour at a call center, defects per unit in manufacturing, website hits per minute, accidents per year at an intersection, and radioactive decay events.
The calculator shows the full PMF histogram, cumulative probabilities, confidence intervals, and compares Poisson to normal approximation. It handles λ values from 0.1 to 200+ and automatically identifies the mode, variance, and skewness.
Model rare event probabilities for quality control, queuing theory, risk assessment, and any scenario where events occur at a known average rate. It is useful when you need both point probabilities and cumulative chances without approximating the process by hand. That makes it practical for staffing, reliability, and event-count analysis.
P(X = k) = (e^(-λ) × λ^k) / k!. Mean = λ. Variance = λ. Standard Deviation = √λ. CDF P(X ≤ k) = Σ P(X = i) for i = 0 to k. Where: λ = average rate (lambda), k = number of events, e = 2.71828...
Result: P(X = 7) = 0.1044, P(X ≤ 7) = 0.8666
With λ = 5, P(X = 7) = e^(-5) × 5^7 / 7! = 0.1044 (10.44%). The cumulative probability P(X ≤ 7) = 0.8666 (86.66%). There's a 13.34% chance of seeing 8 or more events.
A Poisson process has three defining properties: (1) events occur independently, (2) two events cannot occur at the exact same instant, and (3) the average rate is constant over time. Real-world data should be tested against these assumptions before applying the Poisson model.
Overdispersion — when the variance exceeds the mean — is a common violation. In that case, a Negative Binomial distribution is more appropriate.
System reliability: if a server crashes on average once every 30 days, the number of crashes in a year follows Poisson(12). The probability of zero crashes: P(X=0) = e^(-12) ≈ 0.0006%.
Load testing: if an API receives 500 requests/second on average, the probability of receiving more than 550 in any given second uses the Poisson CDF.
When you observe k events and want to estimate the true rate λ, the 95% confidence interval is approximately (k - 1.96√k, k + 1.96√k) for large k, or use exact chi-squared bounds for small k.
Lambda is the expected (average) number of events in the interval. If a store averages 12 customers per hour, λ = 12 for a 1-hour interval, λ = 6 for 30 minutes, or λ = 24 for 2 hours.
Use Poisson when counting events in a continuous interval (time, area, volume) where events are independent and the rate is constant. Use binomial when counting successes in a fixed number of discrete trials.
If events follow a Poisson process with rate λ, the time between events follows an exponential distribution with rate λ. Poisson counts events; exponential measures waiting times.
Yes. λ = 2.5 means an average of 2.5 events per interval. The distribution gives probabilities for 0, 1, 2, 3, ... events. The most likely values are near 2-3.
The normal approximation N(λ, λ) is accurate when λ > 20. The distribution becomes nearly symmetric at large λ, and P(X = k) ≈ normal PDF evaluated at k.
Call center staffing (calls/hour), insurance claim frequency, website server load (requests/second), radioactive decay, typos per page, goals per soccer match, and traffic accidents per intersection per year. It is also useful whenever you need a quick way to estimate the chance of clustered events over a fixed interval.