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-   Calculus & Beyond Homework (http://www.physicsforums.com/forumdisplay.php?f=156)

 hsong9 Sep10-09 08:29 PM

1. The problem statement, all variables and given/known data
Let X0 be a random variable with values in a countable set I. Let Y1, Y2, ... be a
sequence of independent random variables, uniformly distributed on [0, 1].
Suppose we are given a function G : I x [0, 1] -> I
and define inductively for n >= 0, Xn+1 = G(Xn, Yn+1).
Show that (Xn)n>=0 is a Markov chain and
express its transition matrix P in terms of G.

2. Relevant equations

3. The attempt at a solution
I know that I need to show that Xn+1 depends on Xn by checking the condition in the definition of Markov chain, and then
try to find some formula for P(Xn+1 = j | Xn=i) in terms of G.

Actually, my background for Markov chain lacks a little, so I have no how I find some formula for P in terms of G.. How do I handle terms of G?
Anybody give me some hints or answer?

 fi1 Sep5-10 03:06 PM