A Model for Birdwatching and other Chronological Sampling Activities

TL;DR

This paper introduces a new probabilistic model based on random interval graphs to analyze and predict outcomes in chronological sampling activities like birdwatching, extending classical problems like the coupon collector's problem.

Contribution

It develops a novel model for chronological sampling activities using random interval graphs, providing new insights into observation expectations and timing.

Findings

Derived formulas for expected observations before collecting all species

Identified time intervals with the highest likelihood of observing all species

Analyzed probabilities of overlapping observations among multiple species

Abstract

In many real life situations one has $m$ types of random events happening in chronological order within a time interval and one wishes to predict various milestones about these events or their subsets. An example is birdwatching. Suppose we can observe up to $m$ different types of birds during a season. At any moment a bird of type $i$ is observed with some probability. There are many natural questions a birdwatcher may have: how many observations should one expect to perform before recording all types of birds? Is there a time interval where the researcher is most likely to observe all species? Or, what is the likelihood that several species of birds will be observed at overlapping time intervals? Our paper answers these questions using a new model based on random interval graphs. This model is a natural follow up to the famous coupon collector's problem.