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08SimHw2sol_all - IEOR 4404 Simulation Prof Mariana...

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IEOR 4404 Assignment #2 Solutions Simulation 3rd October 2008 Prof. Mariana Olvera-Cravioto Page 1 of ?? Assignment #2 Solutions 1. (a) p X ( x ) = 3 y =0 p ( x, y ) for x = 0,1,2 and 0 otherwise. Thus, p X (0) = 6 30 , p X (1) = 10 30 and p X (2) = 14 30 . p Y ( y ) = 2 x =0 p ( x, y ) for y = 0,1,2,3 and 0 otherwise. Thus, p Y (0) = 3 30 , p Y (1) = 6 30 , p Y (2) = 9 30 and p Y (3) = 12 30 . The plot for p X ( x ) versus x consists of the three points: (0, 6 30 ), (1, 10 30 ) and (2, 14 30 ). Similarly, the plot for p Y ( y ) versus y consists of the four points: (0, 3 30 ), (1, 6 30 ), (2, 9 30 ) and (3, 12 30 ). (b) X and Y are not independent. To see why, simply note that p (0 , 0) = p X (0) × p Y (0). (c) F X ( x ) = 0 , x < 0 6 30 , 0 x < 1 16 30 , 1 x < 2 1 , x 2 F Y ( y ) = 0 , y < 0 3 30 , 0 y < 1 9 30 , 1 y < 2 18 30 , 2 y < 3 1 , x 3 See Figure 1 and 2 on the last page for the plots. (d) E [ X ] = x xP X ( x ) = 38 / 30 E [ X 2 ] = x x 2 P X ( x ) = 66 / 30 Thus, var [ X ] = E [ X 2 ] = E [ X ] 2 = 536 / 900 Similarly, E [ Y ] = y P Y ( y ) = 2 E [ Y 2 ] = y y 2 P Y ( y ) = 5 Thus, var [ Y ] = 1 Finally, cov ( X, Y ) = E [ XY ] - E [ X ] E [ Y ] = x,y xyp ( x, y ) - E [ X ] E [ Y ] = 72 / 30 - 76 / 30 = - 4 / 30 And, corr ( X, Y ) =
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