Module 2:Poisson Process and Kolmorogov equations
  Lecture 5:Posson Process
 

Poisson process
Definition 1: A counting process denoted by  is said to be a Poisson process if it has the following three properties which are:

  1. : Which means that the number of events when the process has just started is zero.

  2. The process has independent increments.

  3. The number of events in the time interval , i.e., of length , where  can be of any length is Poisson distributed with mean . Thus for all values of  and   we have , where  is the rate of the process.

Definition 2 : A counting process denoted by  is said to be a Poisson process with rate   if it has the following four properties which are:

  1. : This means the number of occurrences at time , i.e., when the process has just started is zero.

  2. The process has stationary (time homogeneity) and independent increments.

  3. : Which means that the probability of the number of events in the time interval   being exactly equal to 1 is equal to the product of the rate of the process and the time interval plus some incremental function of time interval, i.e., .

  4. : The fourth property denotes that in case we are interested to find the probability that the number of events in the time interval  is equal to 2 or more, then that probability becomes zero as the time interval shrinks or is made smaller and smaller. For the benefit of the reader we like to mention that a function  is said to be  if we have