My Math Forum Markov Chain problem

 March 5th, 2012, 04:47 PM #1 Senior Member   Joined: Nov 2009 Posts: 169 Thanks: 0 Markov Chain problem Consider the Ehrenfest urn model in which M molecules are distributed among two urns, and at each time point one of the molecules is chosen at random and is then removed from its urn and placed in the other one. Let Xn denote the number of molecules in urn 1 after the nth switch and let $u_n= E[X_n]$. Show that 1) $u_{n+1}= 1+(1-2/M)u_n$ 2) Use (1) to prove that $u_n= \displaystyle \frac{M}{2}+\displaystyle \frac{M-2}{M}^n(E[X_0]-\displaystyle \frac{M}{2})$ I have trouble starting the question, any help will be appreciated!
 March 6th, 2012, 04:17 PM #2 Global Moderator   Joined: May 2007 Posts: 6,756 Thanks: 696 Re: Markov Chain problem There is some ambiguity in the problem statement. What is the process of selecting a molecule at random? Do you first select an urn at random (with what probability) and then select a molecule? Alternatively do you simply select a molecule from either urn, with selection probability for any molecule being the same as any other?

 Tags chain, markov, problem

### consider the ehrenfest urn model in which m molecules are distributed among 2 urns, show that

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