lecture11_random_walks

lecture11_random_walks - COMPSCI 102 Discrete Mathematics...

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COMPSCI 102 Discrete Mathematics for Computer Science
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Probability Refresher What’s a Random Variable? A Random Variable is a real-valued function on a sample space S E[X+Y] = E[X] + E[Y]
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Conditional Expectation What does this mean: E[X | A]? E[ X ] = E[ X | A ] Pr[ A ] + E[ X | A ] Pr[ A ] Pr[ A ] = Pr[ A | B ] Pr[ B ] + Pr[ A | B ] Pr[ B ] Is this true: Yes! Similarly: E[X | A] = ∑ k Pr[X = k | A]
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Random Walks Lecture 12
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How to walk home drunk
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No new ideas Solve HW problem Eat Wait Work Work 0.3 0.3 0.4 0.99 0.01 probability Hungry Abstraction of Student Life
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Abstraction of Student Life Like finite automata, but instead of a determinisic or non-deterministic action, action Example questions: “What is the probability of reaching goal on string Work,Eat,Work?” No new ideas Solve HW problem Eat Wait Work Work 0.3 0.3 0.4 0.99 0.01 Hungry
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- Simpler: Random Walks on Graphs At any node, go to one of the neighbors of the node with equal probability
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- Simpler: Random Walks on Graphs At any node, go to one of the neighbors of the node with equal probability
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- Simpler: Random Walks on Graphs At any node, go to one of the neighbors of the node with equal probability
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- Simpler: Random Walks on Graphs At any node, go to one of the neighbors of the node with equal probability
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- Simpler: Random Walks on Graphs At any node, go to one of the neighbors of the node with equal probability
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0 n k Random Walk on a Line You go into a casino with $k, and at each time step, you bet $1 on a fair game You leave when you are broke or have $n Question 1: what is your expected amount of money at time t? Let X t be a R.V. for the amount of $$$ at time t
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0 n k Random Walk on a Line You go into a casino with $k, and at each time step, you bet $1 on a fair game You leave when you are broke or have $n X t = k + δ 1 + δ 2 + . .. + δ t, ( δ i is RV for change in your money at time i) So, E[X t ] = k E[ δ i ] = 0
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0 n k Random Walk on a Line You go into a casino with $k, and at each time step, you bet $1 on a fair game You leave when you are broke or have $n Question 2: what is the probability that you leave with $n ?
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Random Walk on a Line Question 2: what is the probability that you leave with $n ? E[ X t ] = k E[ X t ] = E[ X t | X t = 0 ] × Pr(X t = 0) + E[ X t | X t = n ] × Pr(X t = n) + E[ X t | neither ] × Pr(neither) Allow play to infinity, but δ i = 0 after reaching 0 or n. As t ∞, Pr(neither) 0, and something t < n Hence Pr(X t = n) E[X t ]/n = k/n = n × Pr(X t = n) + (something t ) × Pr(neither)
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0 n k Another Way To Look At It You go into a casino with $k, and at each time step, you bet $1 on a fair game You leave when you are broke or have $n Question 2: what is the probability that you leave with $n ? = probability that I hit green before I hit red
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- What is chance I reach green before red?
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This document was uploaded on 01/17/2012.

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lecture11_random_walks - COMPSCI 102 Discrete Mathematics...

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