24. Let $ N (t), t 0 % be a Poisson process with rate . Suppose that N (t) is the total number of

24. Let $ N (t), t ≥ 0 % be a Poisson process with rate λ. Suppose that N (t) is the total number of two types of events that have occurred in [0, t]. Let N1(t) and N2(t) be the total number of events of type 1 and events of type 2 that have occurred in [0, t], respectively. If events of type 1 and type 2 occur independently with probabilities p and 1 − p, respectively, prove that $ N1(t), t ≥ 0 % and $ N2(t), t ≥ 0 % are Poisson processes with respective rates λp and λ(1 − p). Hint: First calculate P ! N1(t) = n and N2(t) = m " using the relation P ! N1(t) = n and N2(t) = m " = .∞ i=0 P ! N1(t) = n and N2(t) = m | N (t) = i " P ! N (t) = i " . (This is true because of Theorem 3.4.) Then use the relation P ! N1(t) = n " = .∞ m=0 P ! N1(t) = n and N2(t) = m " .