The Poisson distribution is a one-parameter discrete probability distribution over the non-negative integers, defined by the PMF $e^{-\lambda}\lambda^k/k!$ with rate parameter $\lambda > 0$, whose validity as a PMF and whose expected value $\lambda$ both follow from the Taylor series of $e^{\lambda}$. It arises as the limiting distribution of a large number of weakly dependent or independent trials each with small individual success probability and fixed expected total (the Poisson paradigm/approximation), and is formally shown to be the limit of the binomial distribution as the trial count grows and the success probability shrinks with their product held constant. This distribution belongs to discrete probability theory's toolkit for modeling counts of rare events, and is presented alongside the closely related methodological distinction between a random variable and its distribution (the two are not interchangeable — a distribution is the generative rule, a random variable is a specific instance).
The Poisson distribution is a one-parameter discrete probability distribution over the non-negative integers, defined by the PMF $e^{-\lambda}\lambda^k/k!$ with rate parameter $\lambda > 0$, whose va…