Poisson Distribution Calculator

The Poisson Distribution Calculator computes the probability of observing exactly $$k$$ events in a fixed interval, given an average rate of $$\lambda$$ events per interval. The Poisson distribution is fundamental for modeling rare or random events.

The Poisson probability mass function is:

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

where $$\lambda > 0$$ is the average rate (expected number of events per interval) and $$k = 0, 1, 2, \ldots$$ is the number of observed events.

Mean, Variance, and Standard Deviation

A remarkable property of the Poisson distribution is that the mean and variance are both equal to $$\lambda$$:

$$\mu = \lambda, \quad \sigma^2 = \lambda, \quad \sigma = \sqrt{\lambda}$$

This equality of mean and variance is a defining characteristic — if observed data shows mean significantly different from variance, the Poisson model may not be appropriate (a condition called overdispersion or underdispersion).

Mode

The mode (most probable value) of the Poisson distribution is:

$$\text{Mode} = \lfloor \lambda \rfloor$$

When $$\lambda$$ is an integer, both $$\lambda$$ and $$\lambda - 1$$ are modes (the distribution is bimodal).

Computational Method

For numerical stability, the calculator computes probabilities in log-space using Stirling's approximation for the factorial:

$$\ln(k!) \approx k \ln k - k + \frac{1}{2} \ln\left(\frac{2\pi}{k}\right) + \frac{1}{12k}$$

$$\ln P(X = k) = k \ln \lambda - \lambda - \ln(k!)$$

This approach avoids overflow from computing large factorials and large powers of $$\lambda$$ directly.

Conditions for the Poisson Model

The Poisson distribution is appropriate when: (1) events occur independently, (2) the average rate is constant over the interval, (3) two events cannot occur simultaneously, and (4) the probability of an event in a small sub-interval is proportional to the length of that sub-interval.

Applications

Classic Poisson applications include: the number of phone calls received by a call center per hour, the number of radioactive decays per second, the number of typos per page, the number of accidents at an intersection per year, the number of mutations in a DNA strand, and the number of server requests per minute. The Poisson distribution also arises as a limit of the binomial distribution when $$n$$ is large and $$p$$ is small, with $$\lambda = np$$.