This Numberphile video explores the scenarios of inter-species competition.

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.

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.

Central Limit Theorem is one of the more important concepts in statistics. To put it simply: it explains why that many random processes exhibit normal distribution. Nice visual exploration of this theorem was recently done by 3Blue1Brown. We invite you to watch if you're not yet familiar with the CLT.

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?