information:classification, structural:expansion, information:filtering, structural:summarization, linguistic:translation
Law of rare events

The rate of an event is related to the probability of an event occurring in some small subinterval (of time, space or otherwise). In the case of the Poisson distribution, one assumes that there exists a small enough subinterval for which the probability of an event occurring twice is "negligible". With this assumption one can derive the Poisson distribution from the Binomial one, given only the information of expected number of total events in the whole interval.

Let the total number of events in the whole interval be denoted by λ. {\displaystyle \lambda.} Divide the whole interval into n {\displaystyle n} subintervals I 1, …, I n {\displaystyle I_{1},\dots,I_{n}} of equal size, such that n > λ {\displaystyle n>\lambda } (since we are interested in only very small portions of the interval this assumption is meaningful). This means that the expected number of events in each of the n subintervals is equal to λ / n. {\displaystyle \lambda /n.}

Now we assume that the occurrence of an event in the whole interval can be seen as a sequence of n Bernoulli trials, where the i {\displaystyle i} -th Bernoulli trial corresponds to looking whether an event happens at the subinterval I i {\displaystyle I_{i}} with probability λ / n. {\displaystyle \lambda /n.} The expected number of total events in n {\displaystyle n} such trials would be λ, {\displaystyle \lambda,} the expected number of total events in the whole interval. Hence for each subdivision of the interval we have approximated the occurrence of the event as a Bernoulli process of the form B (n, λ / n). {\displaystyle {\textrm {B}}(n,\lambda /n).} As we have noted before we want to consider only very small subintervals. Therefore, we take the limit as n {\displaystyle n} goes to infinity.

In this case the binomial distribution converges to what is known as the Poisson distribution by the Poisson limit theorem.

In several of the above examples — such as, the number of mutations in a given sequence of DNA—the events being counted are actually the outcomes of discrete trials, and would more precisely be modelled using the binomial distribution, that is X ∼ B (n, p). {\displaystyle X\sim {\textrm {B}}(n,p).}

In such cases n is very large and p is very small (and so the expectation n p is of intermediate magnitude). Then the distribution may be approximated by the less cumbersome Poisson distribution X ∼ Pois (n p). {\displaystyle X\sim {\textrm {Pois}}(np).}

This approximation is sometimes known as the law of rare events, since each of the n individual Bernoulli events rarely occurs.

The name "law of rare events" may be misleading because the total count of success events in a Poisson process need not be rare if the parameter n p is not small. For example, the number of telephone calls to a busy switchboard in one hour follows a Poisson distribution with the events appearing frequent to the operator, but they are rare from the point of view of the average member of the population who is very unlikely to make a call to that switchboard in that hour.

The variance of the binomial distribution is 1 − p times that of the Poisson distribution, so almost equal when p is very small.

The word law is sometimes used as a synonym of probability distribution, and convergence in law means convergence in distribution. Accordingly, the Poisson distribution is sometimes called the "law of small numbers" because it is the probability distribution of the number of occurrences of an event that happens rarely but has very many opportunities to happen. The Law of Small Numbers is a book by Ladislaus Bortkiewicz about the Poisson distribution, published in 1898.