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A sequence of random convex polygons is generated by the following scheme. At each stage, the current polygon is divided into two polygons by choosing two distinct edges at random and joining their midpoints; one of these polygons is cho- sen at random as the current polygon for the next stage. For every choice, each possible outcome is equally likely and all choices are made independently. For n = 0, 1, 2, ..., let X₁ + 3 be the number of edges of the polygon at the nth stage, so that X, takes non-negative integer values. Determine the transition matrix of the Markov chain {Xn, n ≥ 0}. Explain why lim P(X, = i) exists and is independent of the number of edges of n→∞ the initial polygon and determine this limit for each i = 0, 1, 2, … ...

A sequence of random convex polygons is generated by the following scheme. At each stage, the current polygon is divided into two polygons by choosing two distinct edges at random and joining their midpoints; one of these polygons is cho- sen at random as the current polygon for the next stage. For every choice, each possible outcome is equally likely and all choices are made independently. For n = 0, 1, 2, ..., let X₁ + 3 be the number of edges of the polygon at the nth stage, so that X, takes non-negative integer values. Determine the transition matrix of the Markov chain {Xn, n ≥ 0}. Explain why lim P(X, = i) exists and is independent of the number of edges of n→∞ the initial polygon and determine this limit for each i = 0, 1, 2, … ...

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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A sequence of random convex polygons is generated by the following scheme. At each stage, the current polygon is divided into two polygons by choosing two distinct edges at random and joining their midpoints; one of these polygons is cho- sen at random as the current polygon for the next stage. For every choice, each possible outcome is equally likely and all choices are made independently. For n = 0, 1, 2, ..., let X₁ + 3 be the number of edges of the polygon at the nth stage, so that X₁, takes non-negative integer values. Determine the transition matrix of the Markov chain {Xn, n ≥ 0}. Explain why lim P(X, = i) exists and is independent of the number of edges of 14∞ the initial polygon and determine this limit for each i = 0, 1, 2, ....
Transcribed Image Text:A sequence of random convex polygons is generated by the following scheme. At each stage, the current polygon is divided into two polygons by choosing two distinct edges at random and joining their midpoints; one of these polygons is cho- sen at random as the current polygon for the next stage. For every choice, each possible outcome is equally likely and all choices are made independently. For n = 0, 1, 2, ..., let X₁ + 3 be the number of edges of the polygon at the nth stage, so that X₁, takes non-negative integer values. Determine the transition matrix of the Markov chain {Xn, n ≥ 0}. Explain why lim P(X, = i) exists and is independent of the number of edges of 14∞ the initial polygon and determine this limit for each i = 0, 1, 2, ....
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