Derivation of Poisson Process

Let us define , which means that  events occur in time .

Now , means the probability of the number of events in the time frame  being zero is equal to the product of the probabilities that the number of events occurring in time frame  and  each is zero. This is true as the property of independent increments holds true for the counting process and the Poisson process we consider here.

In the conceptual sense this time  is taken to be infinitesimally small, i.e.,  (Figure 2.4).

                                     Figure 2.4: The concept of time h as it becomes zero

Hence

Solving this using simple integration furnishes us with , where  is the constant of integration