Poisson Distribution Calculator

Poisson Distribution Guide

The Poisson Distribution

P(X = k) = (λᵏ × e⁻ˡ) / k!. Where λ = mean number of events per interval, k = number of events, e ≈ 2.71828. The Poisson distribution models events that occur randomly and independently at a constant average rate. Properties: mean = variance = λ. Standard deviation = √λ. The distribution is right-skewed for small λ and approaches the normal distribution as λ increases (above 15-20).

When to Use the Poisson Distribution

Conditions: events occur independently. Events cannot occur simultaneously. The average rate (λ) is constant over the interval. Common applications: number of calls to a call centre per hour (λ = 45 calls/hour). Number of accidents at a junction per month (λ = 2.3). Radioactive decay events per second. Number of bacteria on a microscope slide. Number of printing errors per page. Number of goals in a football match (roughly Poisson, λ ≈ 2.7 in Premier League). Website visitors per minute.

Poisson as Binomial Approximation

The Poisson distribution approximates the binomial when n is large and p is small (rare events). Use Poisson when n ≥ 50 and p ≤ 0.05. λ = np. Example: 1000 items, 0.3% defect rate. n=1000, p=0.003. λ = 3. P(X ≥ 5) using Poisson ≈ P(X ≥ 5) using binomial — much simpler calculation. The approximation is because n!/(r!(n-r)!) becomes unwieldy for large n, while the Poisson formula uses only λ.

Fitting a Poisson Distribution

Test whether data follows a Poisson distribution: calculate the sample mean and variance — if the data is Poisson, these should be approximately equal (mean = variance). Large variance relative to mean (overdispersion) suggests the Poisson model is inappropriate — a negative binomial distribution may fit better. Chi-squared test: compare observed frequencies to expected Poisson frequencies. If χ² < critical value, the Poisson model fits. Application: insurance companies use Poisson models for cl