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 N 1 (t) and N 2 (t) be the total number of events...

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