Poisson limit theorem

The Poisson limit theorem states that a binomial distribution converges to a Poisson distribution as the trial count grows, success probability shrinks, and their product approaches a fixed mean.

Binomial distribution: A discrete probability distribution counting successes in a fixed number of independent trials with the same success probability. The theorem starts with this distribution and takes its rare-event limit.
Convergence in distribution: A mode of convergence in which cumulative distribution functions approach the limiting distribution at its continuity points. The theorem's claim concerns distributional convergence, not equality at finite trial counts.
Rare-event modeling: The statistical modeling of infrequent occurrences across repeated opportunities, time, or space. The theorem justifies Poisson counts when each opportunity has a small event probability.
Normal approximation to the binomial: An approximation of a binomial distribution by a normal distribution when its mean and variance are sufficiently large. Unlike the Poisson limit, this approximation suits many moderate-probability trials.
Poisson distribution: A discrete probability distribution for counts of events with a fixed average rate over a specified interval or region. This is the limiting distribution as binomial trials become numerous and individually unlikely.
Bernoulli distribution: A probability distribution on zero and one, representing a single trial with a specified probability of success. Each binomial trial is Bernoulli, with its success probability shrinking in the limit.
Quality control: Methods for monitoring and improving whether products or processes meet specified requirements. Defect counts among many items can be approximated by a Poisson distribution when defects are rare.
Geometric distribution: A discrete distribution for the number of Bernoulli trials required to obtain the first success. It models waiting time between successes rather than the binomial count taken to a Poisson limit.
Law of rare events: A probabilistic principle that counts of many independent, individually unlikely events often approach a Poisson distribution. This broader principle captures the theorem's rare-event interpretation.
Expected value: The probability-weighted average value of a random variable. The product of trial count and success probability is the binomial mean held fixed.

Linked from 5 pages