The bus stop paradox, or renewal paradox, is a phenomenon in probability theory where the observed average waiting time for an event (such as a bus arrival) may be greater than the actual average waiting time. The paradox arises because the observer is more likely to arrive during a long interval between events than during a short interval.
In more detail, the distribution of the time it takes for the next bus to arrive at a bus stop after our arrival may be very different from the distribution of interval between arrivals of busses at the same bus stop throughout the day. The difference is a combination of the law of our arrival and the distribution of arrivals of buses at the bus stop. Assume we arrive uniformly at random at any point of the day.
If the interval between busses is constant, say a bus every 30 minutes, then the expected waiting time will be of 15 minutes. On the other hand, if the intervall betwenn busses is exponentially distributed, with expected value of 30 minutes, then the expected waiting time will be of 30 minutes.
An intuitive way to explain the difference between the two cases is to say that one is more likelly to come to the bus stop when the interval between busses is the largest. More formally, the resolution of the paradox lies in the fact that the observer's waiting time is not a random variable independent of the bus arrival process but is instead a conditional random variable whose distribution depends on the arrival times of the buses.
We see the above paradox as a basic example to motivate the study of renewal theory. Renewal theory is an essential part of the broader study of stochastic processes, since it serves as a foundation for more advanced topics such as:
Moreover renewal theory offers insights into the long-term behavior of stochastic systems, including the distribution of events over time and the convergence of empirical measures. In this project we can explore different aspects of renewal theory such as
Prerequisites
Probability I.