# Poisson distribution inside-out: estimate the beginning of the period based on first occurrence time

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!

• The time interval between the start of the process and the first occurrence is an "inter-arrival time". For a Poisson distribution, the inter-arrival times are exponentially distributed random variables having mean $\frac{1}{\lambda}$. – pg1989 Aug 29 '17 at 1:41
• Don't conflate the Poisson process with the Poisson distribution; in this case you're looking at inter-event times in the process, which are not Poisson-distributed. Pay attention to the "most likely" part; in particular, you need to take care how you interpret that. – Glen_b Aug 29 '17 at 2:17