Let X denote the number of events in a unit interval of time or in a unit distance. Then, X is called the Poisson random variable with mean number of events λ in a unit interval of time. The probability mass function of a Poisson distribution with mean λ is given by

f(k|λ) = P (X = k|λ) = e −λλ k

k! , k = 0, 1, 2, . . . . (6.1)

The cumulative distribution function of X is given by

F (k|λ) = P (X ≤ k|λ) = k∑ i=0

e−λλ i

i! , k = 0, 1, 2, . . . . (6.2)

The Poisson distribution can also be developed as a limiting distribution of the binomial, in which n → ∞ and p → 0 so that np remains a constant. In other words, for large n and small p, the binomial distribution can be approximated by the Poisson distribution with mean λ = np. Some examples of the Poisson random variable are:

1. the number of radioactive decays over a period of time;

2. the number of automobile accidents per day on a stretch of an interstate road;

3. the number of typographical errors per page in a book;

4. the number of α particles emitted by a radioactive source in a unit of time;

5. the number of still births per week in a large hospital.