Unformatted text preview: ition matrix like the following:
0
1
2
3 0
1/2
1/24
1/36
1/8 1
1 /2
7 /8
0
0 2
3
0
0
1/12
0
8/9 1/12
0
7 /8 The idea behind this matrix is that if left undisturbed a grassy area will see
bushes grow, then small trees, which of course grow into large trees. However,
disturbances such as tree falls or ﬁres can reset the system to state 0. Find the
limiting fraction of land in each of the states.
More Theoretical Exercises
1.45. Consider a general chain with state space S = {1, 2} and write the
transition probability as
1
2
11a
a
2
b
1b
Use the Markov property to show that
P (Xn+1 = 1) b
= (1
a+b a and then conclude
P (Xn = 1) = b
+ (1
a+b a ⇢ b) P (Xn = 1) b
a+b ⇢
b)n P (X0 = 1) b
a+b This shows that if 0 < a + b < 2, then P (Xn = 1) converges exponentially fast
to its limiting value b/(a + b).
1.46. Bernoulli–Laplace model of di↵usion. Consider two urns each of which
contains m balls; b of these 2m balls are black, and the remaining 2m b are
white. We say that the system is in state i...
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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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