Markov Chain problem
- From: isabellesup@xxxxxxxxxxx
- Date: Thu, 09 Aug 2007 12:39:39 -0700
Hi all,
This is the text of a problem I am struggling with:
Suppose an urn contains one black marble and one white marble. One
marble is drawn at random and the color noted. If the color is black,
it is put back in the urn with an extra black marble, and if it is
white, it is put back in the urn with an extra white marble.
This process is repeated, so that at each step an extra marble of the
same color as the one currently drawn is placed in the urn. Let Rn
denote the proportion of black marbles in the urn at the nth step.
Question: Show that the chain R1, R2....Rn has the Markov property:
P{R(n+1)=r |R1....Rn}=P{R(n+1)=r|Rn}
What I did so far is I wrote the network of possible states and how to
get from one 'node' to another. This problem reminds me of the
binomial model.
Further, I was thinking that
Given n and Rn,
1/the probability of drawing a black marble from the urn is
[Rn/(n+1)] (at the first step, n=1, this proba equals 1/2)
2/the probability of drawing a white marble from the urn is
[(n+1-Rn)/(n+1)] (at the first step, n=1, this proba equals 1/2)
So that
P{R(n+1)=r |R1....Rn}=
P{R(n)=r | no black marble is drawn}
+P{R(n)=r-1 | a black marble is drawn}
=P{R(n)=r,no black marble is drawn}/[R(n)/(n+1)] + P{R(n)=r-1,a black
marble is drawn}/[(n+1-R(n))/(n+1)]
I am not too sure where to go from here...
Your help is greatly appreciated.
.
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