Show that if {Ni (t ), t 0} are independent Poisson processes with rate i ,

Show that if {Ni (t ), t ≥ 0} are independent Poisson processes with rate λi , i = 1, 2, then {N(t), t ≥ 0} is a Poisson process with rate λ1 + λ2 where N(t) = N1(t)+ N2(t).

41. Let {N(t), t ≥ 0} be a Poisson process with rate λ.

(a) Find P(N(4) = 4|N(3) = 1).

(b) Find Var(N(8)|N(5) = 6).

(c) Find P(N(5) = 0|N(8) −N(3) = 4).