May 2, 2014

Stick Breaking

Problem Set 1: Given a stick of length 1 meter. It is cut into two parts randomly.
  • Problem 1a: What is the expected length of the smaller stick?
  • Problem 1b: What is the expected length of the bigger stick?

Problem Set 2: Given a stick of length 1 meter. It is cut into three parts randomly.
  • Problem 2a: What is the expected length of the smallest stick?
  • Problem 2b: What is the expected length of the largest stick?
  • Problem 2c: What is the expected length of the stick with second smallest (or second largest) length?
  • Problem 2d: What is the expected length of the middle stick (stick that is between the two cuts)?
  • Problem 2e: What is the probability that the three sticks would form a triangle?
  • Problem 2f: What is the probability that the three sticks would form a right angle triangle?
  • Problem 2g: What is the probability that the three sticks would form an equilateral triangle?

Problem Set 3: Given a stick of length 1 meter. It is cut into two parts randomly. Then bigger stick is picked and is again cut into two parts randomly.
  • Problem 3a: What is the expected length of the smallest stick?
  • Problem 3b: What is the expected length of the largest stick?
  • Problem 3c: What is the expected length of the stick with second smallest (or second largest) length?
  • Problem 3d: What is the expected length of the middle stick (stick that is between the two cuts)?
  • Problem 3e: What is the probability that the three sticks would form a triangle?
  • Problem 3f: What is the probability that the three sticks would form a right angle triangle?
  • Problem 3g: What is the probability that the three sticks would form an equilateral triangle?

Generalized Cut Problem: Find the solution to problem 2a-2c, 3a-3c for n-1 cuts.

Solution: show

No comments:

Post a Comment