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# 01_30ans - &amp;amp&amp;amp-=-3 1 2 2 3 1 1 2 2 1...

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STT 410 Examples for 01/30/2008 (Answers) Spring 2008 1. x \ y 0 1 2 1 0.15 0.15 0 0.30 2 0.15 0.35 0.20 0.70 0.30 0.50 0.20 1.00 a) P ( X + Y = 2 ) = p ( 1, 1 ) + p ( 2, 0 ) = 0.15 + 0.15 = 0.30 . b) P ( X > Y ) = p ( 1, 0 ) + p ( 2, 0 ) + p ( 2, 1 ) = 0.15 + 0.15 + 0.35 = 0.65 . c) x p X ( x ) y p Y ( y ) 1 2 0.30 0.70 0 1 2 0.30 0.50 0.20 d) E ( X ) = 1 × 0.30 + 2 × 0.70 = 1.7 . E ( Y ) = 0 × 0.30 + 1 × 0.50 + 2 × 0.20 = 0.9 . E ( X + Y ) = 1 × 0.15 + 2 × 0.15 + 2 × 0.15 + 3 × 0.35 + 3 × 0 + 4 × 0.20 = 2.6 . OR E ( X + Y ) = E ( X ) + E ( Y ) = 1.7 + 0.9 = 2.6 . E ( X Y ) = 0 × 0.15 + 0 × 0.15 + 1 × 0.15 + 2 × 0.35 + 2 × 0 + 4 × 0.20 = 1.65 . 2. ( ) ° ± ² + = otherwise 0 1 , 1 0 , 1 0 24 , y x y x y x y x f a) 1. f ( x , y ) 0 for all ( x , y ) . ° 2. ( ) ( ) ( ) ³ ³ ³ ³ ³ - = ´ ´ µ · · ¸ ¹ = - - - 1 0 2 1 0 1 0 1 12 24 , dx x x dx dy y x dy dx y x f x ( ) ( ) 0 1 3 8 6 12 24 12 4 3 2 1 0 3 2 x x x dx x x x + - = + - = ³ = 1. °

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b) P ( X + Y < 0.5 ) = ( ) ( ) ³ ³ ³ - = ´ ´ µ · · ¸ ¹ - 5 . 0 0 2 5 . 0 0 5 . 0 0 5 . 0 12 24 dx x x dx dy y x x ( ) 0 5 . 0 3 4 2 3 12 12 3 4 3 2 5 . 0 0 3 2 ´ ´ µ · · ¸ ¹ + - = + - = ³ x x x dx x x x 16 1 = + - = 16 3 2 1 8 3 = 0.0625 . c) P ( 2 X Y ) = ( ) ( ) [ ] ( ) ³ ³ ³ - - = ´ ´ µ · · ¸ ¹ - 3 1 0 2 2 3 1 0 1 2 2
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Unformatted text preview: &amp;amp; &amp;amp;--= -3 1 2 2 3 1 1 2 2 1 12 24 dx x x x dx dy y x x x ( ) ( ) 3 1 9 8 6 36 24 12 4 3 2 3 1 3 2 x x x dx x x x +-=--= &amp;amp; 27 7 = +-= 81 9 27 8 9 6 0.25926 . d) If x [ , 1 ], f X ( x ) = 0. If x [ , 1 ], f X ( x ) = ( ) 2 1 1 12 24 x x dy y x x-= &amp;amp;-. e) E ( X ) = ( ) ( ) &amp;amp; &amp;amp; +-=- 1 4 3 2 1 2 12 24 12 1 12 dx x x x dx x x x = ( ) 1 2.4 6 4 5 4 3 x x x +-= 0.40. By symmetry, f Y ( y ) = ( ) 2 1 12 y y-for y [ , 1 ], f Y ( y ) = 0 for y [ , 1 ]. E ( Y ) = 0.40. Total cost = ( 1.00 ) X + ( 1.50 ) Y + ( 0.50 ) ( 1 X Y ) = 0.5 + 0.5 X + Y. E ( Total cost ) = 0.5 + 0.5 E ( X ) + E ( Y ) = \$1.10 ....
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