Now by mathematical indication we can show that

Remark

(1) The parameter .  is called the rate or intensity of the Poisson process. It can be shown that

Thus the probability of the increments depends on the length of the interval and not on the absolute times

The Poisson process is a process with stationary increments.

(2) The independent and stationary increment properties help us to compute the joint probability mass function of

For example,