In a homogeneous Poisson process with events that occur at a rate of \(\lambda\) per unit time, we now define a discrete random variable, \(X\), to be the number of events that occur in one unit of time.

If the unit of time is split into infinitessimally small periods, each of length \(\delta t\), the requirements of a homogeneous Poisson process mean that

  1. No more than one event can occur in each period.
  2. The occurrences of events in different small periods are independent.
  3. The probability of an event in any interval is \(\lambda \times \delta t\).

Binomial approximation

To derive the probability function of \(X\), we start with a situation in which the above three conditions hold, but \(\delta t\) is larger. If unit time is split into \(n\) intervals, each of length \(\frac 1 n\), the three conditions would mean that the number of events is the number of successes in a series of \(n\) independent Bernoulli trials, each with probability \(\pi = \frac {\lambda} n\) of success, so

\[ X \;\sim\; \BinomDistn \left(n, \pi = \frac {\lambda} n\right) \]

The probability function for the number of events in a homogeneous Poisson process can be found as the limit of the probability function of this binomial distribution, as \(n \to \infty\).

Poisson probability function

The number of events in unit time in a Poisson process with rate \(\lambda\) per unit time has probability function

\[ p(x) \;\;=\;\; \frac {\lambda^x e^{-\lambda}} {x!} \quad\quad \text{ for } x=0, 1, \dots \]

(Proved in full version)

A distribution of this form is called a Poisson distribution.