IE410%20PP%20SLIDES%20CH1

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

This preview shows pages 1–9. Sign up to view the full content.

(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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
(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 .
(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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
(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).
(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)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
(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).
(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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
(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).
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 90

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

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online