Number of regions created by random chords in the circle

TL;DR

This paper investigates the distribution of the number of regions formed by randomly drawn chords in a circle, showing it converges to a normal distribution as the number of chords increases, with bounds on the approximation error.

Contribution

It establishes a Central Limit Theorem for the number of regions created by random chords and provides explicit error bounds using Stein's method.

Findings

Number of regions converges to a normal distribution as n increases.

Explicit bounds on the approximation error are derived.

The distribution's behavior is characterized for large n.

Abstract

In this paper we discuss the number of regions in a unit circle after drawing $n$ i.i.d. random chords in the circle according to a particular family of distribution. We find that as $n$ goes to infinity, the distribution of the number of regions, properly shifted and scaled, converges to the standard normal distribution and the error can be bounded by Stein's method for proving Central Limit Theorem.