# chapter4 - 1 Chapter 4 41(a x 0 1 pX(x 27/50 11/50 y 0 1...

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1 Chapter 4 4–1. (a) x 0 1 2 3 4 5 p X ( x ) 27/50 11/50 6/50 3/50 2/50 1/50 y 0 1 2 3 4 p Y ( y ) 20/50 15/50 10/50 4/50 1/50 (b) y 0 1 2 3 4 p Y | 0 ( y ) 11/27 8/27 4/27 3/27 1/27 (c) x 0 1 2 3 4 5 p X | 0 ( x ) 11/20 4/20 2/20 1/20 1/20 1/20 4–2. (a) y 1 2 3 4 5 6 7 8 9 p Y ( y ) 44/100 26/100 12/100 8/100 4/100 2/100 2/100 1/100 1/100 x 1 2 3 4 5 6 p X ( x ) 26/100 21/100 17/100 15/100 11/100 10/100 (b) y 1 2 3 4 5 6 7 8 9 x = 1 p Y | 1 ( y ) 10/26 6/26 3/26 2/26 1/26 1/26 1/26 1/26 1/26 x = 2 p Y | 2 ( y ) 8/21 5/21 3/21 2/21 1/21 1/21 1/21 0 0 x = 3 p Y | 3 ( y ) 8/17 5/17 2/17 1/17 1/17 0 0 0 0 x = 4 p Y | 4 ( y ) 7/15 4/15 2/15 1/15 1/15 0 0 0 0 x = 5 p Y | 5 ( y ) 6/11 3/11 1/11 1/11 0 0 0 0 0 x = 6 p Y | 6 ( y ) 5/10 3/10 1/10 1/10 0 0 0 0 0 4–3. (a) Z 10 0 Z 100 0 k 1000 dx 1 dx 2 = 1 k = 1 (b) f X 1 ( x 1 ) = Z 10 0 1 1000 dx 2 = 1 100 ; 0 x 1 100 = 0; otherwise f X 2 ( x 2 ) = Z 100 0 1 1000 dx 1 = 1 1000 x 1 100 0 = 1 10 ; 0 x 2 10 = 0; otherwise

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2 (c) F X 1 ,X 2 ( x 1 ,x 2 ) = 0; x 1 0 ,x 2 0 = 0; x 1 0 ,x 2 > 0 = 0; x 1 > 0 ,x 2 0 = Z x 1 0 Z x 2 0 1 1000 dt 2 dt 1 = x 1 x 2 1000 ;0 < x 1 < 100 , 0 < x 2 < 10 = Z x 1 0 Z 10 0 dt 2 dt 1 1000 = x 1 100 ;0 < x 1 < 100 ,x 2 10 = Z x 2 0 Z 100 0 dt 1 dt 2 1000 = x 2 10 ; x 1 100 , 0 < x 2 < 10 = 1; x 1 100 ,x 2 10 . 4–4. (a) k Z 4 2 Z 2 0 (6 - x 1 - x 2 ) dx 1 dx 2 = 1 k = 1 8 (b) P ( X 1 < 1 ,X 2 < 3) = Z 3 2 Z 1 0 1 8 (6 - x 1 - x 2 ) dx 1 dx 2 = 3 8 (c) P ( X 1 + X 2 4) = 1 8 Z 4 2 Z 4 - x 2 0 (6 - x 1 - x 2 ) dx 1 dx 2 = 2 3 (d) f X 1 ( x 1 ) = 1 8 Z 4 2 (6 - x 1 - x 2 ) dx 2 = 1 8 (6 - 2 x 1 );0 x 1 2 P ( X 1 < 1 . 5) = Z 3 / 2 0 1 8 (6 - 2 x 1 ) dx 1 . = 0 . 844 (e) f X 1 ( x 1 ) see (d) f X 2 ( x 2 ) = Z 2 0 1 8 (6 - x 1 - x 2 ) dx 1 = 1 8 (10 - 2 x 2 );2 x 2 4 4–5. (a) P ± W 2 3 ,Y 1 2 = Z 2 / 3 0 Z 1 0 Z 1 / 2 0 Z 1 0 16 wxyz dz dy dxdw = 1 9 (b) P ± X 1 2 ,Z 1 4 = Z 1 0 Z 1 / 2 0 Z 1 0 Z 1 / 4 0 16 wxyz dz dy dxdw = 1 64 (c) Z 1 0 Z 1 0 Z 1 0 16 wxyz dz dy dx = 2 w ;0 w 1 = 0; otherwise Note: x , y , z and w are independent
3 4–6. From Problem 4–4 f X ( x ) = 1 8 (6 - 2 x );0 x 2 f Y ( y ) = 1 8 (10 - 2 y ); 2 y 4 f X | y ( y ) = f X,Y ( x,y ) f Y ( y ) = 6 - x - y 10 - 2 y ;0 x 2 , 2 y 4 f Y | x ( x ) = f X,Y ( x,y ) f X ( x ) = 6 - x - y 6 - 2 x ;2 y 4 , 0 x 2 4–7. E ( Y | X = 3) = 9 X y =1 y · p Y | 3 ( y ) = 33 17 4–8. (a) x 1 51 52 53 54 55 p X 1 ( x 1 ) 0.28 0.28 0.22 0.09 0.13 x 2 51 52 53 54 55 p X 2 ( x 2 ) 0.18 0.15 0.35 0.12 0.20 (b) x 2 51 52 53 54 55 otherwise P X 2 | 51 ( x 2 ) 6 28 7 28 5 28 5 28 5 28 0 P X 2 | 52 ( x 2 ) 5 28 5 28 10 28 2 28 6 28 0 P X 2 | 53 ( x 2 ) 5 22 1 22 10 22 1 22

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chapter4 - 1 Chapter 4 41(a x 0 1 pX(x 27/50 11/50 y 0 1...

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