Intuition behind Poisson process

Question

I have the following problem in my book which is testing the intuition behind poisson process:

If cars arrive according to a poisson process, what are you able to say about the arrival process of the passengers in the car?

This question is very strange to me. I don't know if you can say anything about the passengers. I just know that the time between the cars will be exponential. How would someone answer this question?

Answer 1

I'm not exactly sure what the question is driving at but I think I have some idea. A Poisson process is memoryless in the sense that knowledge of past events cannot inform the probability of future events occurring. The arrival of the cars, as stated by the problem, is a such process. The arrival of each passenger is however correlated—they tend to arrive in groups!

So if for instance a passenger has arrived in the last minute, there is some probability $x$ that at least another will arrive in the next minute. If on the other hand no passengers have arrived in the past minute, there is some probability $y$ that one or more passengers will arrive in the next minute, and without even knowing the probabilities we can still confidently claim that $x>y$.

Answer 2

The passengers arrive according to a compound Poisson process. Essentially, a compound Poisson process's jumps still arrive according to a typical Poisson process, but the jump sizes are allowed to be random. In your case, the jump times correspond to car arrivals, and the jump sizes correspond to the number of passengers in a car.