Last time we have introduced a concept of waiting time. We have assumed that some time has passed since the last event and wanted to how long do we have to wait until another event. The time since the last event was a thing we knew. Now let us consider a slightly different case - what happens when we do not know when the last event happened, how long do we have to wait until the next event?

In other words, imagine you just have come to a bus stop. If you know that the average bus interarrival time is \( 15 \) minutes, then how long would you expect to have to wait? Straightforward answer seems to be \( 7.5 \) minutes, but is it a correct answer?

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.