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516hw10

# 516hw10 - Section 5.1 1 a 7/12 4 b 5/36 4 1 1 1 2 0 0 1 2 2...

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Unformatted text preview: Section 5.1 1. a) 7/12 4 b) 5/36. 4 1 1 1 2 0 0 1 2 2. a) 0.1 b) 1 - 2(1/ 2)(0.19) 2 = 0.0975 (0.2) 2 0.2 0.01 0.01 4. a) 1 - ( 3 / 4 ) = 7 /16 = 0.4375 2 0.2 1 0.25 0 0.25 1 X b) P - 1 Y 4 0.25 P X = 5 Y 4 1 4 3 X 1 - + 9 / 40 = 0.225 = = 3 2 5 4 3 4 1 4 5 c) P (Y X | Y 0 1 1/ 2 - 1/ 32 5 0.25) = = = 0.625 3/ 4 8 1 1 4 0 1 5. a) 0.1 b) P (| X - Y |> 0.1) = (9 /10) 2 = 0.81 1 0.1 0 0.1 6. a) (1/2)132/152 = 0.376 1 Jill 15 13 2 15 Jack b) Let A be the event that the first person arrives before 12:05 and let B be the event that the last person arrives after 12:10. Then P ( AB ) = 1 - P ( Ac B c ) = 1 - P ( Ac ) - P( B c ) + P ( Ac B c ) = 1 - (10 /15)10 - (10 /15)10 + (5 /15)10 = 0.965 Section 5.2 2. a) f XY ( x, y ) = 1/ 2 for | x | + | y |< 1 and 0 otherwise . 1-| x| b) If |x|<1 then f X ( x ) = - (1-| x|) (1/ 2)dy = 1- | x | . Otherwise, f X ( x ) =0. Density of Y would be exactly the same for symmetry reasons. c) Not independent as f XY ( x, y ) P f X ( x) fY ( y ) . d) E(X) = E(Y) = 0. 3. a) c = . 3 3 a 2 b) + a 4 3 c) 3 b 2 +b 4 3 -2 x -3 y 4. a) ( 1 - e ) ( 1 - e ) b) 2e -2 x for x > 0. c) 3e -3 y for y > 0 d) They are independent as the marginal densities multiply to the joint density. - x- y dx = dy 5. e + 3 0 3y 7. Without loss of generality the circle has radius 1. Then P ( R P r ) = r 2 . So the density is given by 2r for 0 < r < 1. This gives the desired probability to be r1 / 2 11 3 4r1r2 dr2 1 = dr1 = 1/8. dr 2 0 r1 0 0 1 ...
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