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classLec2 - IEOR 4404 Simulation Lecture 2 January 23rd...

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IEOR 4404 Simulation Lecture 2 January 23rd, 2012 Mariana Olvera-Cravioto [email protected] 1

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Probabilistic models Definition: An experiment is a process whose outcome is not known with certainty. Definition: The set of possible outcomes of an experiment is called the sample space , which we will denote S . The outcomes themselves are called sample points . Examples: Flipping a coin ⇒ S = { H, T } Tossing a 6-sided die ⇒ S = { 1 , 2 , 3 , 4 , 5 , 6 } Flipping a coin 10 times S = { all “words” of length 10 that have letters H and T only } The time it will take for your call to be picked up by an airline’s call center ⇒ S = [0 , ) The score in the next Knicks game S = { ( x, y ) : x, y are nonnegative integers } IEOR 4404, Lecture 2 2
Probability laws A probability law P assigns to each event A ⊆ S a value in [0 , 1]. Let Ω = S be the universe and Ø denote the empty set. Axioms: Let A, B Ω P ( A ) 0 If A B = Ø, then P ( A B ) = P ( A ) + P ( B ) P (Ω) = 1 Other properties: Let A, B Ω If A B , then P ( A ) P ( B ) P ( A B ) = P ( A ) + P ( B ) - P ( A B ) P (Ø) = 0 IEOR 4404, Lecture 2 3

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Independence and conditional probabilities Events A and B are independent if P ( A B ) = P ( A ) P ( B ) For any events A and B in the sample space, with P ( B ) > 0, the conditional probability of A given B is defined as P ( A | B ) = P ( A B ) P ( B ) Conditional probabilities specify a probability law. IEOR 4404, Lecture 2 4
Random Variables and their properties Definition: A random variable is a function that assigns a real number to each point in the sample space. X : S → R Examples: Tossing a 6-sided die: X = result of the toss, X ∈ { 1 , 2 , 3 , 4 , 5 , 6 } The weather tomorrow: let X = ( 1 , if it rains 0 , if it does not rain. , X ∈ { 0 , 1 } Let X be the time you will have to wait for the subway next time you take it. X [0 , ) Let X be the Knicks’ score in their next game and let Y be that of their opponent. Let Z = ( X, Y ) be the overall score of the game. Z ∈ { ( x, y ) : x, y are nonnegative integers } IEOR 4404, Lecture 2 5

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Indicator random variables In general, for any event A , the random variable X = ( 1 , if A happens 0 , if A does not happen ( A c happens) is known as an “indicator” random variable.
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