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Zahra_Mohaghegh_April 29th_Markov

Zahra_Mohaghegh_April 29th_Markov - Engineering Management...

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1 Engineering Management & Systems Engineering The George Washington University EMSE 102 EMSE 102 - - 202 202 Survey of Operations Research Methods Survey of Operations Research Methods Zahra Mohaghegh [email protected] Post-Doctoral Research Associate Center For Risk and Reliability University of Maryland April 29, 2008

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2 Engineering Management & Systems Engineering The George Washington University Markov Chains Markov Chains CREDITS: PARTS OF THIS LECTURE HAVE BEEN DEVELOPED BY PROFESSORS MOSLEH (UMD) AND CAMPOS- NONEZ (GWU)
3 Engineering Management & Systems Engineering The George Washington University Where we are Where we are Linear and Nonlinear modeling Integer and Noninteger modeling Static and Dynamic modeling Deterministic and Stochastic modeling A deterministic model is a model in which for any value of the decision variables the value of the objective function and whether or not the constraints are satisfied is known with certainty. If this is not the case, then we have a stochastic model . Random Variable The study of how random variable changes over time includes Stochastic Processes A type of stochastic process is Markov Chain

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4 Engineering Management & Systems Engineering The George Washington University Review of Random Variable (1) Review of Random Variable (1) Definition : A random variable is a function that associates a number with each point in an experiment’s sample space. We denote random variables by boldface capital letters (usually X, Y, or Z ).
5 Engineering Management & Systems Engineering The George Washington University Review of Random Variable (2) Review of Random Variable (2) Definition : A random variable is discrete if it can assume only discrete values x 1 , x 2 ,…. A discrete random variable X is characterized by the fact that we know the probability that X = x i (written P( X =x i )). P( X =x i ) is the probability mass function (pmf) for the random variable X .

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6 Engineering Management & Systems Engineering The George Washington University Review of Random Variable (3) Review of Random Variable (3) If, for some interval, the random variable X can assume all values on the interval, then X is a continuous random variable . Probability statements about a continuous random variable X require knowing X ’s probability density function (pdf).
7 Engineering Management & Systems Engineering The George Washington University Classification Classification

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8 Engineering Management & Systems Engineering The George Washington University Stochastic Process Stochastic Process
9 Engineering Management & Systems Engineering The George Washington University Example: Discrete Example: Discrete - - time Stochastic Process time Stochastic Process

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10 Engineering Management & Systems Engineering The George Washington University Example: Continuous Example: Continuous - - time Stochastic time Stochastic Process Process
11 Engineering Management & Systems Engineering The George Washington University Classification Classification

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12 Engineering Management & Systems Engineering The George Washington University A Counting Process A Counting Process
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Zahra_Mohaghegh_April 29th_Markov - Engineering Management...

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