Where the Poisson distribution applies; the axiomatic approach to probability

(i) Situations for the Poisson distribution

The Poisson distribution, P(x) = e⁻ᵐ mˣ / x!, describes the number of times a rare event happens in a fixed interval of time, space or quantity, where m is the average number in the interval and is also the variance. It can be used when:

1. the event is rare: the chance of it happening in any small part of the interval is very small, while the number of opportunities for it is very large;
2. two or more happenings in a very small part of the interval are practically impossible;
3. happenings are independent of each other and of time, occurring at random at a steady average rate; and
4. as a limit of the binomial, n is large, p is small and np = m is moderate.

Examples.

• Number of customers' complaints a call centre receives in an hour.
• Number of defective items in a large batch from a process that is usually well controlled.
• Number of typing errors on a page of a long report.
• Number of accidents at a road junction in a week, or of insurance claims an office receives in a day.
• Number of breakdowns of a machine in a month.

(ii) The axiomatic approach to probability

The classical approach needs equally likely outcomes and the relative-frequency approach needs a long run of trials. The axiomatic (modern) approach, due to A. N. Kolmogorov (1933), avoids both by defining probability through three axioms only. For a sample space S, a function P that gives each event A a number P(A) is a probability if:

1. Non-negativity: P(A) ≥ 0 for every event A;
2. Certainty: P(S) = 1 for the sure event; and
3. Additivity: for mutually exclusive events A₁, A₂, A₃, …, P(A₁ ∪ A₂ ∪ A₃ ∪ …) = P(A₁) + P(A₂) + P(A₃) + ….

Every other rule follows from these: P(∅) = 0, P(A′) = 1 − P(A), 0 ≤ P(A) ≤ 1, and the general addition theorem P(A ∪ B) = P(A) + P(B) − P(A ∩ B). The approach does not say how the numbers are to be found; the classical or the empirical method can supply them, and the axioms guarantee that the results behave consistently. Its merit is that it covers every case, including outcomes that are not equally likely and experiments that cannot be repeated.