If a Fisherman catches his first fish of the day at 10am, when is he more likely to have started fishing given that in average he catches 1 fish/hour?
In other words: Let's assume a Poisson process with given (known) lambda and starting at an unknown moment in time. Given that the first occurrence happened exactly at 10am, when the Poisson process is most likely to have started? Is it possible to have confidence intervals?
My intuition says sometime before 10am but I cannot get a formal answer... although I feel I need to go Bayesian!