IE410%20PP%20SLIDES%20CH1

IE410%20PP%20SLIDES%20CH1 - CS481/IE410 STOCHASTIC...

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(C) 2009 Jacobson (C) 2009 Jacobson STOCHASTIC MODELS STOCHASTIC MODELS 1 CS481/IE410 STOCHASTIC PROCESSES AND ITS APPLICATIONS CHAPTER 1. Introduction to Probability Theory 1.1: WHAT IS PROBABILITY? A philosophy • A way of looking at the world Outcome is not predictable in advance Examples: - events that are safe and those that are risky, - flying on a plane, riding in a car, playing the lottery, gambling, investing
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(C) 2009 Jacobson (C) 2009 Jacobson STOCHASTIC MODELS STOCHASTIC MODELS 2 1.2: DEFINITIONS A random experiment is an experiment where Outcomes are not predictable in advance. • There are several possible outcomes. The set of all possible outcomes of a random experiment, S, is called the sample space .
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(C) 2009 Jacobson (C) 2009 Jacobson STOCHASTIC MODELS STOCHASTIC MODELS 3 EXAMPLE 1: A machine produces sheets of metal. Each output piece is inspected and then classified as either good {G} or waste {W}. Therefore, S = {G, W}. Suppose that some wasted sheets can be reworked {R}. Then S = {G, R, W}. Suppose that there are two machines working on the job shop floor. Then S = {GG, GR, GW, RG, RR, RW, WG, WR, WW}. Output of machine 1 Output of machine 2
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(C) 2009 Jacobson (C) 2009 Jacobson STOCHASTIC MODELS STOCHASTIC MODELS 4 EXAMPLE 2: A sales call lasts a certain amount of time. Then S = [0,+ ). A sales call lasts no more than 60 minutes. Then S = [0,60]. A sales call lasts between 45 and 75 minutes. Then S = [45,75]. A sales call lasts less than 30 minutes. Then S = [0,30).
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(C) 2009 Jacobson (C) 2009 Jacobson STOCHASTIC MODELS STOCHASTIC MODELS 5 In the Sequential Stochastic Assignment game that you played, S = All possible sums of products of numbers 1 to 6. In a perfect world, where all numbers appear exactly once, Smallest number in S: 56 = 2*(1*6 + 2*5 + 3*4) Largest possible number: 91 = 1 2 + 2 2 + 3 2 + 4 2 + 5 2 + 6 2 Question: Are all numbers between 56 and 91 possible? Can we get 90? 90 = 1 2 + 2 2 + 3 2 + 4 2 + 5*6 + 6*5 We would need to enumerate all possible combinations! In a non-perfect world, the range of numbers is 21 (all 1’s) to 126 (all 6’s)
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(C) 2009 Jacobson (C) 2009 Jacobson STOCHASTIC MODELS STOCHASTIC MODELS 6 Subsets of S are called events . In EXAMPLE 1, the event of at least one sheet being good is E = {GG, GR, GW, RG, WG}. The event of no sheet being good is F = {RW, WR, RR, WW}. In EXAMPLE 2, the event that a call lasts more than 4 minutes but less than 6 minutes is E = (4,6).
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(C) 2009 Jacobson (C) 2009 Jacobson STOCHASTIC MODELS STOCHASTIC MODELS 7 Operations can be performed on events. UNION: E F are elements in S that are in E or F or both. INTERSECTION: E F = EF are elements in S that are in both E & F. COMPLEMENT: E c are elements in S that are not in E. E F E F S
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(C) 2009 Jacobson (C) 2009 Jacobson STOCHASTIC MODELS STOCHASTIC MODELS 8 CONTAIN: E F means that all elements of E are in F. Note: In some books, means contains in (including equality), while means strictly contains in (does not include equality).
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IE410%20PP%20SLIDES%20CH1 - CS481/IE410 STOCHASTIC...

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