> Even if the bus system was poorly run, why would it make sense to assume that the expected value of time to arrival doesn't change based on how long you've been waiting?
I don't think it's saying anything about how long you've been waiting, and you don't know when was the last arrival.
It's saying that if you pick a random point on the timeline, the expected wait time doesn't change. That's because by taking a random point you have more chances of landing in a larger stretch of wait time than in a smaller one.
> I don't think it's saying anything about how long you've been waiting, and you don't know when was the last arrival.
This is exactly what memorylessness says something about.
Your second paragraph isn't unique to Poisson processes, but the author right at the start says that the expected value of the waiting time is the same as the average interarrival time, which indicates Poisson.
I don't think it's saying anything about how long you've been waiting, and you don't know when was the last arrival.
It's saying that if you pick a random point on the timeline, the expected wait time doesn't change. That's because by taking a random point you have more chances of landing in a larger stretch of wait time than in a smaller one.