The memorylessness of the Poisson process makes the statistical aspect a bit trivial. But here's an interesting variant: how should you update your beliefs while waiting if there is a certain probability that the bus won't come at all? "The Ups and the Downs of the Hope Function in a Fruitless Search", Falk et al 1994: https://www.gwern.net/docs/statistics/bayes/1994-falk