Few important and interesting properties of Poisson process

Property 2.1

• Suppose  is a Poisson process with rate . Furthermore assume that the probability
  of an event being recorded is  and  is the number of recorded events in . Then
   is a Poisson process, i.e., , with rate .

Proof of property 2.1

Property 2.2

• Suppose  is a Poisson process with rate . Furthermore consider events are of
  two types which are say
  Type I (male) and Type II (female), with corresponding probabilities of  and .
  Thus we have  
  and here  is of Type I while  is of Type II.
  Then both  and  will be Poisson process with rates
   and  respectively and they would be independent.
  
  

Proof of property 2.2