What Distribution Is The Coat Hangers Problem

What Distribution is the Coat Hanger Problem? A Deep Dive into Probability

The "coat hanger problem," also known as the "broken stick problem" or variations thereof, is a classic probability puzzle that explores the distribution of lengths when a stick is broken into multiple pieces. While the exact phrasing varies, the core question remains: what's the probability distribution of the lengths of the resulting pieces? The answer isn't straightforward and depends heavily on how the stick is broken. This article will explore different scenarios and the resulting distributions.

Understanding the Problem's Variations:

The problem's ambiguity lies in the method of breaking the stick. Several interpretations exist:

• Uniformly at Random: The stick is broken at two points chosen independently and uniformly at random along its length. This is the most common interpretation and leads to a specific distribution.
• Breaking into Two Pieces: The stick is broken only once, at a point chosen uniformly at random. This simplifies the problem considerably.
• Specific Break Points: The break points are predetermined or follow a specific non-uniform distribution.

The Uniformly Random Case (Two Breaks):

This is the most challenging variant. Let's say the stick has length 1. We break it at two points, X and Y, chosen uniformly and independently from the interval [0, 1]. We are interested in the distribution of the three lengths: X, Y-X, and 1-Y (assuming X<Y). This problem reveals a surprising result.

To analyze this, we need to consider the joint distribution of X and Y. Since they are chosen uniformly and independently, the joint probability density function (PDF) is simply f(x,y) = 1 for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1. The lengths of the three pieces are X, Y-X, and 1-Y. The distribution of these lengths is not uniform. It's a more complex distribution related to the Beta distribution. Specifically, the distribution of the lengths is characterized by a particular case of the Dirichlet distribution, which is a generalization of the Beta distribution to more than two variables.

The Single Break Case:

Breaking the stick once at a point chosen uniformly at random is far simpler. The lengths of the two pieces will follow a uniform distribution between 0 and the length of the original stick. If the stick has length L, then the length of one piece will follow a uniform distribution U(0, L).

Implications and Applications:

Understanding the distribution of lengths in the coat hanger problem has applications beyond simple mathematical puzzles. Similar principles apply in various fields, including:

• Material Science: Modeling the fracture of materials.
• Queueing Theory: Analyzing the lengths of waiting times.
• Computational Geometry: Dealing with randomly generated segments.

Conclusion:

The distribution in the coat hanger problem isn't a single, simple answer. The nature of the distribution critically depends on the method of breaking the stick. While the single-break case leads to a straightforward uniform distribution, the more common two-break case yields a more intricate distribution related to the Beta and Dirichlet distributions, highlighting the complexity of seemingly simple probability problems. Further exploration requires delving into multivariate probability distributions and their properties. Remember to always clarify the assumptions and method of breaking when tackling variations of this classic probability problem.