Suppose that {N (t) : t 0} is a Poisson process with rate > 0.

(a) If t > s provide an expression for P ( N(t) = n | N(s) = m ) , consider the cases n m and n < m separately.

(b) Again assume that t > s, provide an expression for cov ( N(t) , N(s) ) = E ( N(t) N(s) ) E ( N (t) ) * E ( N (s) ).

Tip: E( N(t) ) = t and Var ( N(t) ) = E( N(t)^2 ) (E( N(t) ))^2 = t.