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Unformatted text preview: IEOR 4404 Simulation Lecture 2 January 23rd, 2012 Mariana OlveraCravioto [email protected] 1 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 6sided 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 ) ≥ • 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 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 6sided die: X = result of the toss, X ∈ { 1 , 2 , 3 , 4 , 5 , 6 } • The weather tomorrow: let X = ( 1 , if it rains , if it does not rain. , X ∈ { , 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....
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This note was uploaded on 03/18/2012 for the course IEOR 3402 taught by Professor Vananhtruong during the Spring '12 term at Columbia.
 Spring '12
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