Let \(N(t)\) be a Poisson process with intensity λ. For each occurrence, we flip a coin: if heads comes up we label the occurrence green, if tails comes up we label it red. The coin flips are independent and \(p\) is the probability to see heads.

1.  Show that the green occurrence form a Poisson process with intensity λp.
2. Connect this with example 2.2.5 from Meester.
3. We claim that the red occurrences on the one hand, and the green occurrences on the other hand form independent Poisson processes. Can you formulate this formally, and prove it , using Example 2.2.5 from Meester once more?

[Meester ex. 7.5.7]