Real arrival streams are rarely a single undifferentiated Poisson process — cars split into trucks and passenger vehicles, customers split by what they buy, defaults come with random loss severities attached. This note gives the three basic operations for building richer models from Poisson processes: compounding (attach a random mark to each arrival and sum the marks), thinning (split one process into several by classifying each arrival), and superposition (merge several processes into one). The genuinely surprising fact, proved below, is that thinning a Poisson process by an independent random label produces independent Poisson processes — seeing an unusually large count in one category tells you nothing about the count in another.
Learning objectives. By the end of this note you should be able to:
- compute the mean and variance of a compound Poisson sum;
- state and apply the thinning theorem, including its remarkable independence conclusion;
- state and apply the superposition theorem;
- use Poisson races (thinning viewed as classification) to compute probabilities like "which type of arrival happens first."
In this lesson
- Compound Poisson processes
- Thinning
- Superposition
The full lesson, with worked examples and exercise solutions, is for members.
Read the full lessonNew to SelfMadeQuant.com? See membership options or explore the curriculum.