Let us suppose that events are indeed occurring at certain(random) points of time, and let us assume that for some positive constant the following assumptions hold true:
Loosely put, assumptions 1 and 2 state that for small values of h, the probability that exactly 1 event occurs in an interval of size h equals plus something that is small compared to h, whereas the probability that 2 or more events occur is small compared to h. Assumption 3 states that whatever occurs in one interval has no (probability) effect on what will occur in other nonoverlapping intervals.
Under assumptions 1,2, and 3, we shall now show that the number of events occurring in any interval of length t is a Poisson random variable with parameter . To be precise, let us call the interval [0,t] and denote by N(t) the number of events occurring in that interval. To obtain an expression for , we start by breaking the interval [0,t] into n nonoverlapping subintervals each of length t/n as follow.
This follows because the event on the left side of Equation (8.2), that is, , is clearly equal to the union of the two mutually exclusive events on the right side of the equation. Letting A and B denote the two mutually exclusive events on the right side of Equation (8.2), we have
Now, for any t, as and so as by the definition of o(h). Hence
On the other hand, since assumptions 1 and 2 imply that
we see from the independence assumption, number 3, that
Thus, from Equations (8.2), (8.3), (8.4), we obtain, by letting ,
Hence, if assumptions 1,2, and 3 are satisfied, the number of events occurring
in any fixed interval of length t is a Poisson random variable with mean
and we say that the events occur in accordance with a Poisson
process having rate .
The value ,
which can be shown to equal
the rate per unit time at which events occur, is a constant that must be