Any matrix with properties i and ii gives rise to a

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Unformatted text preview: Xn . 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 long-run 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.

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