Stochastic

# A what is the expected number of steps that xn will

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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.

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