markov_practice_problems

# markov_practice_problems - state from any other...

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Some practice problems involving Markov Chains December 20, 2010 1. Show that if T is a Markov chain such that there is a positive probability of transitioning from any vertex to any other vertex in a single time step (note: we are not saying that there is a positive probability of transitioning from vertex i to itself in one time step), then T has a unique equilibrium distirubiton. 2. Suppose there is an epidemic in which every month half of those who are well become sick, and a quarter of those who are sick become well. Find the steady state for the corresponding Markov process. 3. Suppose that an ant starts at position (0 , 0), and cane move up, down, left, or right one unit, so long as the new position says inside the grid { ( x, y ) : 0 x, y 2 } . Determine the probability that the ant reaches position (2 , 2) within 20 moves. 4. Find an example of a connected Markov chain – you can reach any

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Unformatted text preview: state from any other state (ignoring edge directions) – that has more than one equilibrium distribution. 5. Population dynamics can often be modeled by composing polynomials, and in this problem we will explore this possibility: suppose that you have a population of bacteria whereby from one generation to the next, a single individual will produce j o±spring with probability p j , and that that individual dies in the next generation. Let f ( x ) = ∞ s j =0 p j x j . 1 Say that in our case f ( x ) = 1 / 2 + 1 / 4 x + 1 / 4 x 2 . If at time 0 the population has 1 individual, determine the expected number of individuals for when time t = 4. Hint: think about what f ( f ( x )) means, and then f ( f ( f ( x ))), etc. 2...
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markov_practice_problems - state from any other...

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