- #1
hsong9
- 80
- 1
Please Help! Markov Chain
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.
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?
Homework Statement
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.
Homework Equations
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?