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Week02-Lecture (20100831)

# Week02-Lecture (20100831) - Simulation Modeling and...

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1 Simulation Modeling and Analysis (ORIE 4580/5580/5581) Week 2: Review of Probability and Statistics (08/31/10 - 09/02/10)

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Announcement and Agenda Recitations begin this Friday (9/3) There will be no recitation on Monday (9/6) because of Labor Day. You can attend the Friday sections, or just review the recitation questions and solutions that will be posted on Blackboard. HW1 is now available on Blackboard: Due Date: 11am on 9/9 Please use the Discussion Board for Homework Questions. I’ve already posted additional hints on one of the threads. Ticket has been filed about the AC in B14. Will keep you posted. Simulation Job Opportunity at Intel Cornell INFORMS Chapter Announcement 2
Sample Space Suppose we perform a random experiment whose outcome cannot be predicted in advance. Sample Space = the set of all possible outcomes of the experiment Example1: The experiment is tossing a die Sample Space = {1, 2, 3, 4, 5, 6} Example 2: The experiment is flipping two coins simulataneously Sample Space = { HH, HT, TH, TT } 3

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Random Variable (RV) Random Variable (RV) is a function that maps the sample space to real numbers. The value of the random variable is determined by the outcome of the random experiment. We typically denote the random variables by capital letters, for example, X, Y, or Z Example 1: The experiment is tossing a die X = 1 / (outcome of the die) Y = outcome of the die + 2 Example 2: The experiment is flipping 2 coins simultaneously X = # of tails 4 Y = 1 , if the two coins are the same , 0 , otherwise.
Discrete Random Variables and Probability Mass Function Definition: Random variables that take only discrete values. Our examples in the previous slide are discrete random variables Suppose we have a discrete random variable X taking integer values p(x) = Pr{ X = x } is called the probability mass function of X NOTE: The distinction between the random variable X (in upper case) and the variable x (in lower case) Properties: p(x) 0 for all x and Example: X = # of tails when we flip 2 fair coins simultaneously. p(0) = 1/4, p(1) = 1/2, p(2) = 1/4, and p(x) = 0 for all x {0, 1, 2} 5 x = −∞ p ( x ) = 1

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Discrete RV: Cumulative Distribution Functions An alternative way to describe the the probability law of a random variable X is to consider its cumulative distribution F( ) 6 F ( x ) = Pr { X x } = x i = −∞ Pr { X = i } = x i = −∞ p ( i ) Knowing F( ) is the same as knowing p( ) because p ( x ) = x i = −∞ p ( i ) x 1 i = −∞ p ( i ) = F ( x ) F ( x 1) Properties of F( ): F ( x ) 0 for all x lim x →−∞ F ( x ) = 0 and lim x →∞ F ( x ) = 1 Pr { a X b } = b i = a p ( i ) = b i = −∞ p ( i ) a 1 i = −∞ p ( i ) = F ( b ) F ( a 1)
Continuous Random Variables Consider the following random variable Y Y = distance between your upper and lower eyelids at the end of the lecture Y can takes any value in a continuum (the positive real line), so it is called a continuous random variable How would we describe the behavior of X?

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