Lecture3 - IEOR 4404 Simulation Lecture 3 January 25th,...

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Unformatted text preview: IEOR 4404 Simulation Lecture 3 January 25th, 2012 Mariana Olvera-Cravioto molvera@ieor.columbia.edu 1 Jointly discrete random variables When analyzing probabilities involving more than one random variable the relationship between them must be specified through their joint distribution . If X and Y are jointly discrete random variables, then p ( x,y ) = P ( X = x,Y = y ) is called the joint PMF of X and Y . If X and Y are independent, then p ( x,y ) = p X ( x ) p Y ( y ) where p X ( x ) = P ( X = x ) and p Y ( y ) = P ( Y = y ). IEOR 4404, Lecture 3 2 Jointly discrete random variables If X and Y are not independent, then we need to fully specify p ( x,y ) for each combination of x and y . Example: Consider tossing two identical 6-sided dice. Let X be the outcome of first die and Y be the sum of the two dice. X { 1 , 2 , 3 , 4 , 5 , 6 } and Y { 2 , 3 , 4 ,..., 11 , 12 } The joint PMF of X and Y , p ( x,y ) is given below: X ,Y 2 3 4 5 6 7 8 9 10 11 12 1 1 36 1 36 1 36 1 36 1 36 1 36 2 1 36 1 36 1 36 1 36 1 36 1 36 3 1 36 1 36 1 36 1 36 1 36 1 36 4 1 36 1 36 1 36 1 36 1 36 1 36 5 1 36 1 36 1 36 1 36 1 36 1 36 6 1 36 1 36 1 36 1 36 1 36 1 36 IEOR 4404, Lecture 3 3 Marginal and conditional PMFs The joint PMF of X and Y contains all the information about X and Y . We can recover the marginal PMFs of X and Y through the formulas p X ( x ) = X y p ( x,y ) and p Y ( y ) = X x p ( x,y ) p ( x,y ) can also be determined using the conditional PMFs : p X | Y ( x | y ) = P ( X = x | Y = y ) = P ( X = x,Y = y ) P ( Y = y ) p Y | X ( y | x ) = P ( Y = y | X = x ) = P ( X = x,Y = y ) P ( X = x ) and the relation p ( x,y ) = p X | Y ( x | y ) p Y ( y ) = p Y | X ( y | x ) p X ( x ) IEOR 4404, Lecture 3 4 Example The number of customers arriving to a certain coffee shop between 8:00 am and 10:00 am is modeled as a Poisson r.v. with parameterand 10:00 am is modeled as a Poisson r....
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This note was uploaded on 03/18/2012 for the course IEOR 4404 taught by Professor C during the Spring '10 term at Columbia.

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Lecture3 - IEOR 4404 Simulation Lecture 3 January 25th,...

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