True/False: Occurrence Times, Counting Process and Waiting Times

Let *S*_{1},*S*_{2},⋯ be the occurrence times of a homogeneous Poisson Process with counting Process *N*(*t*) and waiting times *X*_{1},*X*_{2},⋯ . True or False:

a) For any i<j, X_{j} is independent of S_{i}

b) For any i<j, X_{i }is independent of S_{j }

c) For any i≠j, S_{i}is independent of S_{j }

d) For any T > S_{i}, N(T) - N(S_{i}) = 0 if and only if S_{i+1 }> T

Where *T* is our index set of *t*, essentially *T* is the total number of events

Re: True/False: Occurrence Times, Counting Process and Waiting Times

Hey Rorschach27.

In terms of independence, we define this as P(A=a,B=b) = P(A=a)*P(B=b). In other words, if you have the joint distribution and can separate it in terms of this product, then you have proven that both processes are independent.