Poisson Distribution Calculator

The Poisson distribution relies on a single parameter: lambda ($Ε$), which represents the average rate of success or expected number of occurrences in a given interval. The probability of observing exactly $x$ events is defined by the probability mass function: P(X = x) = (Ε^x * e^-Ε) / x!, where $e$ is Euler's number approximately equal to 2.71828. Cumulative probabilities evaluate the likelihood of seeing a range of outcomes, such as fewer than or greater than a specific threshold value, by summing the individual point probabilities.

Imagine a local coffee shop that averages 4 customer arrivals every 10 minutes (Ε = 4). You want to find the exact probability that exactly 2 customers arrive in the next 10-minute window ($x = 2$). Plugging these values into the formula: P(X = 2) = (4^2 * e^-4) / 2! = (16 * 0.018315) / 2 = 0.29304 / 2 = 0.1465. Thus, there is a 14.65% chance that exactly 2 customers will arrive. The calculator also automatically computes cumulative metrics, such as the chance of 2 or fewer arrivals.

Ensure that your events are independent of one another; the occurrence of one event must not alter the probability of another. Verify that your chosen rate of success (Ε) is constant and directly proportional to the length of the observation window. Avoid using this model if events can happen simultaneously or if the population size is small and sampling is done without replacement.