Unformatted text preview: in the denominator, (ii) the indices decrease by 1 each time, (iii) the answer
will not depend on q` (which is 0) or p`+i .
For a concrete example to illustrate the use of this formula consider
Example 1.31. Ehrenfest chain. For concreteness, suppose there are three
balls. In this case the transition probability is
0
1
2
3 0
1
2
3
0
3 /3
0
0
1/3
0
2 /3
0
0
2/3
0
1/3
0
0
3 /3
0 Setting ⇡ (0) = c and using (1.12) we have
⇡ (1) = 3c, ⇡ (2) = ⇡ (1) = 3c ⇡ (3) = ⇡ (2)/3 = c. 33 1.6. SPECIAL EXAMPLES
The sum of the ⇡ ’s is 8c, so we pick c = 1/8 to get
⇡ (0) = 1/8, ⇡ (1) = 3/8, ⇡ (2) = 3/8, ⇡ (3) = 1/8 Knowing the answer, one can look at the last equation and see that ⇡ represents the distribution of the number of Heads when we ﬂip three coins, then
guess and verify that in general that the binomial distribution with p = 1/2 is
the stationary distribution:
⇡ (x) = 2
Here m! = 1 · 2 · · · (m n ✓◆
n
x 1) · m, with 0! = 1, and ✓◆
n
n!
=
x!(n x)!
x is the binomial coe cient which gives the number of ways of choosing x objects
out of a set of n.
To check that our...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.
 Spring '10
 DURRETT
 The Land

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