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To construct the chain we can think of playing a board game. When we are in
state i, we roll a die (or generate a random number on a computer) to pick the
next state, going to j with probability p(i, j ).
Example 1.3. Weather chain. Let Xn be the weather on day n in Ithaca,
NY, which we assume is either: 1 = rainy, or 2 = sunny. Even though the
weather is not exactly a Markov chain, we can propose a Markov chain model
for the weather by writing down a transition probability
1
.6
.2 1
2 2
.4
.8 The table says, for example, the probability a rainy day (state 1) is followed by
a sunny day (state 2) is p(1, 2) = 0.4. A typical question of interest is:
Q. What is the longrun fraction of days that are sunny?
Example 1.4. Social mobility. Let Xn be a family’s social class in the nth
generation, which we assume is either 1 = lower, 2 = middle, or 3 = upper. In
our simple version of sociology, changes of status are a Markov chain with the
following transition probability
1
2
3 1
.7
.3
.2 2
.2
.5
.4 3
.1
.2
.4 Q. Do the fractions of people in the thre...
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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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