Consider a random process representing the number of occurrences of an event up to time t (over time interval ). Such a process is called a counting process and we shall denote it by

Clearly is a continuous-time discrete state process and any of its realizations is non-decreasing function of time.

The counting process is called Poisson's process with the rate parameter if

1. N (0)=0
2. N ( t ) is an independent increment process.

Figure 1

Thus in Figure 1, the increments