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Topics_covered - Introductory Engineering Stochastic...

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Introductory Engineering Stochastic Processes, ORIE 361 Instructor: Mark E. Lewis, Associate Professor School of Operations Research and Information Engineering Cornell University Spring, 2008 1/ 28
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Disclaimer This file can be used as a study guide. Please note that as the semester progresses, some of the material may be adjusted (added to or subtracted). This is due to the fact that the class may require further clarification on some topics and less on others depending on your strengths and weaknesses. 2/ 28
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Preliminaries 1 Course Prerequisites 1 Basic knowledge of random variables Discrete random variables Continuous random variables 2 Independence 3 Expectation Functions of random variables (Law of the unconscious statistician) Expectation of linear combinations of random variables Variance of linear combinations of independent random variables 4 Moment Generating Functions (mgf’s) k th moments via differentiating mgf’s Mgf of sums of independent random variables Uniqueness of mgf’s (equal mgf’s equal distributions) 5 Conditional probability and expectation 3/ 28
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Markov chain (DTMCs) 2 Define a stochastic process A sequence of random variables An outcome is called a sample path The set of possible outcomes is called the state space , say S . 3 The Markov Property Given the present, the future is independent of the past When the state space is discrete P ( X n +1 = j | X n = i , X n - 1 = i n - 1 , . . . , X 0 = i 0 ) = P ( X n +1 = j | X n = i ) . for all i , j , i n - 1 , . . . i 0 S . You can think of this as a refinement of independence; the above would be LHS = P ( X n +1 = j ). A discrete time stochastic process with the Markov property is called a discrete-time Markov chain (DTMC) . Since p ij = P ( X n +1 = j | X n = i ) is defined for all i , j S , this defines a matrix P (with ( i , j ) th element p ij ). The p ij ’s are independent of n ; this is called time homogeneity . 4/ 28
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Transient distributions; P ( X n = j | X 0 = i ) = p ( n ) ij Chapman-Kolmogorov equations p n + m ij = X k S p ( n ) ik p ( m ) kj = ( P n P m ) ij (just multiply matrices) Initial distributions: Let α j = P ( X 0 = j ), then P ( X n = k ) = X j S α j p ( n ) jk (condition on the initial state) 5/ 28
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Long-Run Behavior; as n → ∞ Need to classify states: starting in state i 1 Transient: return to state i a finite number of times 2 Recurrent: return to state i an infinite number of times More formally: let T i = min { n > 0 | X n = i } 1 Transient: P ( T i < ∞| X 0 = i ) < 1 2 Recurrent: P ( T i < ∞| X 0 = i ) = 1 Several checks: let f i = probability of starting in i we ever return to state i 1 Transient: f i < 1. Alternatively, n =0 p ( n ) ii < 2 Recurrent: f i = 1. Alternatively, n =0 p ( n ) ii = j is accessible from i (written i j , not a limit ) if p ( n ) ij > 0 for some n 0 (note “=” is included) j communicates with i if i j and i j (written i j ) Communication is an equivalence relation The state space can be divided into disjoint equivalence classes If all states in the state space communicate, the DTMC is said to be irreducible 6/ 28
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Long-Run Behavior (2) Theorem If state i is recurrent (transient) and i j, then j is recurrent (transient).
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