Waiting time distribution

Last time we have seen that interarrival (or, more generally, inter-event) times in the Poisson process follow exponential distribution. Inter-event times tell us how much time has passed since the last event, but we are often also interested in times till event given that \( T \) time has passed since previous event.

In the terms of the original problem we could ask the question: what is the expected time for the next student to come? Let us assume that \( 5 \) minutes has passed since the arrival of the last student. Let us recall that \( 4 \) students arrive per hour (meaning on average \( 15 \) minutes between them). Intuitive and wrong answer would be \( 10 \) minutes.

Poisson process: Interarrival times

Last time we have built a quick model for independent student arrival. We have assumed that there is some probability that a student will arrive during some short time interval. Then we have taken the continuum limit, and forgot the microscopic model. In this post let us take a look what was left behind the scenes in the last: interarrival time distribution.

Poisson process

Let us assume that you are a college professor. You teach and introductory course, so effectively you have infinitely many students. Any of them can have a chat with you right after the lunch on Friday. Lets say your office hours start around 1 p.m., and end at around 2 p.m.. The problem is that your students are completely unpredictable! But after long years of teaching you have figured out that on average you meet \( 4 \) students each Friday. With \( 95 \% \) confidence what is the maximum number of students that will come to see you?