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Problem 1 (Ehrenfest's Diffusion Model). Let N be a container separated into the left ar he right parts by a barrier in the middle. The container is filled with K particles in tota At each time n = 1,2, · . ·, we randomly pick one particle among the K particles and pla t into the other part of the container. Let {Xn}nɛN be the stochastic process where X he number of particles in the left part of the container at time n. The state space of ti process is therefore X = {0,1, ·. . , K}. %3D i) Explain why {X„}n€N is a Markov process.

Problem 1 (Ehrenfest's Diffusion Model). Let N be a container separated into the left ar he right parts by a barrier in the middle. The container is filled with K particles in tota At each time n = 1,2, · . ·, we randomly pick one particle among the K particles and pla t into the other part of the container. Let {Xn}nɛN be the stochastic process where X he number of particles in the left part of the container at time n. The state space of ti process is therefore X = {0,1, ·. . , K}. %3D i) Explain why {X„}n€N is a Markov process.

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Problem 1 (Ehrenfest's Diffusion Model). Let N be a container separated into the left and the right parts by a barrier in the middle. The container is filled with K particles in total. At each time n = 1, 2, · .., we randomly pick one particle among the K particles and place it into the other part of the container. Let {Xn}n€N be the stochastic process where X, is the number of particles in the left part of the container at time n. The state space of the process is therefore X = {0, 1, · .. , (i) Explain why {Xn}nɛN is a Markov process. (ii) Write down the transition probability matrix P for {Xn}nɛN and analyze its invariant distribution.
Transcribed Image Text:Problem 1 (Ehrenfest's Diffusion Model). Let N be a container separated into the left and the right parts by a barrier in the middle. The container is filled with K particles in total. At each time n = 1, 2, · .., we randomly pick one particle among the K particles and place it into the other part of the container. Let {Xn}n€N be the stochastic process where X, is the number of particles in the left part of the container at time n. The state space of the process is therefore X = {0, 1, · .. , (i) Explain why {Xn}nɛN is a Markov process. (ii) Write down the transition probability matrix P for {Xn}nɛN and analyze its invariant distribution.
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