How long do you need to wait for the bus?
Suppose there is a bus, 🚌, it arrives at the bus stop every 10 minutes, how long, on average, do you need to wait for the bus?
Let random variable X represent waiting time. X follows a uniform distribution. In extreme cases, the individual either waits 10 minutes (when arrives, the bus just left) or no time, (just in time to catch the bus).
Hence, we have a = 0, b = 10 for uniform distribution. The mean of the uniform distribution is 1/2*(a+b), we have 5 mins.
However, if, on average, the bus arrives at the bus stop every 10 minutes, how long, on average, do you need to wait for the bus?
I chose to write this blog is because of this interesting phenomenon.
The number of bus arrivals in a certain time period is considered a Poisson distribution. The time interval between bus arrivals then follows an Exponential distribution. One interesting property of Exponential distribution is memoryless i.e. P(T>t+s|T>s) = P(T>t). The literal explanation is that, when you arrive at the bus stop, it doesn’t matter how long the next arriving bus has traveled. As a result, the waiting time is the expected arrival interval, which is 10 minutes in this case.
This is known as the Waiting Time Paradox. I am still looking for a more intuitive explanation of the second question. For people who are interested, please visit https://jakevdp.github.io/blog/2018/09/13/waiting-time-paradox/
