Poisson Distribution Calculator

What Is the Poisson Distribution?

The Poisson distribution models the count of rare, independent events over a fixed interval when the average rate is known. It is widely used for arrivals, defects, and event counts. Related tools: Geometric Distribution Calculator and Binomial Probability Distribution Calculator.

The probability mass function is:

$$P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!}$$

Where $\lambda$ is the average rate of occurrences and $x$ is a non-negative integer.

For a Poisson distribution, the mean and variance both equal $\lambda$, so the standard deviation is $\sqrt{\lambda}$.

When should I use the Poisson distribution?

Use it when events are independent, occur at a roughly constant average rate, and two events cannot happen at the exact same instant.

What is lambda in this calculator?

Lambda ($\lambda$) is the expected number of occurrences in the interval you are analyzing.

How is Poisson related to the binomial distribution?

Poisson approximates binomial counts when $n$ is large and $p$ is small while $np$ stays moderate.

Can x be negative or non-integer?

No. The Poisson random variable counts events, so x must be a non-negative integer.