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10SimHw2 - IEOR 4404 Simulation Prof Mariana...

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IEOR 4404 Assignment #2 Simulation February 4, 2010 Prof. Mariana Olvera-Cravioto Page 1 of 2 Assignment #2 – due February 12th, 2010 1. For any random variables X 1 , X 2 and any numbers c 1 , c 2 , show that var( c 1 X 1 + c 2 X 2 ) = c 2 1 var( X 1 ) + 2 c 1 c 2 cov( X 1 , X 2 ) + c 2 2 var( X 2 ) . 2. There are 7 traffic lights on your way to work. The free flow time (when you hit greens on all lights) is 15 minutes. Upon approaching each light, there is independent probability of 20% that it will be red. Each time you encounter a red light, the delay time is a random variable X , where X is assumed to be uniformly distributed in (0.5, 2) minutes. Find the mean and variance of your commute time. 3. Dependence and Correlation. We want to examine the degree of dependence between two events and the relation of this dependence to a measure of correlation. For two events A and B we define the indicator random variables X and Y as follows: X = ( 1 , if A occurs, 0 , otherwise , Y = ( 1 , if B occurs, 0 , otherwise. Assume 0 < P ( A ) , P ( B ) < 1. (a) Compute the correlation coefficient of X and Y , ρ ( X, Y ), in terms of P ( A ), P ( B ), and P ( A | B ). ρ ( X, Y ) = cov( X, Y ) p var( X )var( Y ) (b) Show that
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