Britannica Sources used in this answer

Why are the mean and variance equal in the Poisson model of radioactive particle emissions?

Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

sample space for a pair of dice — probability theory Sample space for a pair of dice.

In the Poisson distribution, the mean and variance are both equal to $μ t$ because the model is derived as a limiting case of the binomial distribution.1

When modeling radioactive decay, the time scale is divided into a large number of very small, independent intervals. In each interval, the probability of emitting a particle is proportional to the length of the interval, while the probability of emitting two or more particles is negligible. As the size of these intervals shrinks toward zero and the number of trials becomes infinite, the binomial distribution converges to the Poisson distribution.1

Because this process assumes independent events occurring at a constant average rate, the mathematical structure of the resulting Poisson distribution inherently dictates that its expected value (mean) and its variance are identical.12