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# 08SimHw2 - IEOR 4404 Simulation Prof Mariana...

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IEOR 4404 Assignment #2 Simulation September 17, 2008 Prof. Mariana Olvera-Cravioto Page 1 of 2 Assignment #2 – due September 23rd, 2008 1. Suppose that X and Y are jointly discrete random variables with p ( x, y ) = x + y 30 , for x = 0 , 1 , 2 and y = 0 , 1 , 2 , 3 0 , otherwise. (a) Compute and plot p X ( x ) and p Y ( y ). (b) Are X and Y independent? (c) Compute and plot F X ( x ) and F Y ( y ). (d) Compute E [ X ], var( X ), E [ Y ], var( Y ), cov( X, Y ) and corr( X, Y ). 2. A particle’s velocity V is normally distributed with mean 0 and variance σ 2 . The particle’s energy is given by W = m V 2 2 , where m > 0 is a constant. (a) What is E [ W ] in terms of m and σ ? (b) What is the PDF of W ? 3. If X and Y have a joint probability density function specified by f ( x, y ) = 2 e - ( x +2 y ) , 0 < x < , 0 < y < find P ( X < Y ). 4. 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 ) . 5. 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%

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08SimHw2 - IEOR 4404 Simulation Prof Mariana...

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