Question

Let {?(?),?≥0} be the counting process for a Poisson process with rate ?. Calculate the temporal covariance, Cov(?(?),?(?+?)).

Answer 1

By, the independent increment property of the Poisson process, the two random variables ?(?) and ?(?+?) - ?(?)) are independent. Thus,  Cov(?(?), ?(?+?) - ?(?)) = 0

Now,

Cov(?(?),?(?+?)) = Cov(?(?), ?(?+?) - ?(?) + ?(?))

=  Cov(?(?), ?(?+?) - ?(?)) +  Cov(?(?), ?(?))

= 0 + Cov(?(?), ?(?))

= Var(N(t))

= $\lambda$ t Since, N(t) ~ Poisson($\lambda$t)