APME_7 - Click to edit Master subtitle style Applied...

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Unformatted text preview: Click to edit Master subtitle style Applied Probability Methods for Engineers Slide Set 7 Click to edit Master subtitle style Chapter 17 of Winston’s Operations Research Book Markov Chains Stochastic Processes n Suppose we observe some measurement in a system at time points 0, 1, 2, … n Let Xt be a random variable for the measurement at time t n Discrete-time stochastic process describes relation among variables X0, X1, X2, … n Random variable Xt may depend on all prior values X0, …, Xt-1, or may depend on some subset of these (or may not depend on any of them at all) Gambler’s Ruin n At time 0 you have $2; at times 1, 2, … you bet $1 and with probability p you win $1 (with probability 1 – p you lose) n Your goal is to get to $4 and if you do, you quit n Let Xt be your position at time t n X0 = 2 n X1 = 3 with probability p and X1 = 1 with probability 1 – p n If Xt = 4, then Xt+i = 4 for all i ≥ 1 Urn Problem n Urn contains two unpainted balls n Choose a ball at random and flip a coin n If ball is unpainted and coin lands on heads, paint the ball red n If ball is unpainted and coin lands on tails, paint the ball black n If ball is painted already, we change its color n Define t as time after tth coin toss n Let [u r b] denote the state of the system, where u is # of painted balls, r is # red, and b is # blue n X0= [2 0 0] n After first toss, we get X1 = [1 1 0] or X1 = [1 0 1] n We can define allowable state transitions, e.g., if Xt = [0 2 0] then Xt+1 = [0 1 1] Stochastic Processes n A stochastic process is characterized by a set of allowable states and probabilities of moving from state to state (which are state dependent) n A continuous time stochastic process the state can be observed at any time point, and may change state at any time point n Example: evolution of stock prices over time Markov Chains n A discrete-time Markov chain is a special type of process in which the state at time t depends only on the state at time t – 1 (and not on the state at times t – 2, t – 3, …) n For t = 0, 1, 2, … n P(Xt+1 = it+1| Xt = it, Xt-1 = it-1, …, X1 = i1, X0 = i0) = P(Xt+1 = it+1| Xt = it) n We will also assume that P(Xt+1 = j| Xt = i) does not depend on t, i.e., P(Xt+1 = j| Xt = i) = pij n The probability of transitioning from state i to j does not depend on t Markov Chains n The terms pij are transition probabilities (from state i to state j) n We call these stationary Markov chains because of our assumption that transition probabilities are not time dependent n In addition to knowing pij we also need to know the probability of being in state i at time zero, denoted by qi n q = [q1 q2 …qs] is the initial probability distribution for the state at time 0 Markov Chains...
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This note was uploaded on 10/27/2010 for the course ESI 6321 taught by Professor Josephgeunes during the Spring '07 term at University of Florida.

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APME_7 - Click to edit Master subtitle style Applied...

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