p chapter 8 solutions 3rd ed. fall 2007

# p chapter 8 solutions 3rd ed. fall 2007 - Probability Third...

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Sheet1 Probability, Third Edition By David J Carr & Michael A Gauger Published by BPP Professional Education Solutions to practice questions h Chapter 8 Solution 8.1 The required probability is: (XY 3)fXY, (0,1) +f , (0,2) +fXY(0,3) +f , (1,1) +fXY(1,2) +f , (2,1) Pr += Err:520 XY , XY , XY 0.7 0.7 Solution 8.2 The marginal probability function of X is: fXY , () Err:509 0,1 +fXY 0, 2 +f , (0, 3 )=0.05 +0.09 +0.11 =0.25 fX(0) =. 0, y , ()XY all y f (1) = fXY () Err:509 1,1 +fXY 1, 2 +f , () 1, 3 =0.18 +0.14 +0.03 =0.35 X S, 1, y XY , () XY all y fXY , () Err:509 2,1 +fXY 2, 2 +f , () 2, 3 =0.13 +0.08 +0.04 =0.25 fX(2) =. 2, y , () XY all y Page 1

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Sheet1 X(3) =SfXY , 3, y =fXY 3,1 +f , ( ) +fXY 3,3 =0.07 +0.06 +0.02 f () , () XY 3,2 , () 0.15 all y Solution 8.3 The marginal probability function of Y given that X=2 is: f , (2,1) 0.13 XY f (1 X 2) = Err:520 Err:520 0.52 Y fX(2) 0.25 fXY(2,2) 0.08 f (2 X 2) = Err:520 ,= 0.32 Y fX(2) 0.25 fXY(2,3) 0.04 fY(3 2) 0.16 Page 2
Sheet1 X== , f fX(2) 0.25 Probability Solutions to practice questions h Chapter 8 Solution 8.4 We are given that the probability density function is: 12 () t ,t =c for 0 tt Err:510 2 fTT, 12 12 The region 0 ===T1 T2 in the TT2 21× plane is a triangle with area 2 T2 (the shaded area in the figure on the right). 2 Since the double integral of the joint pdf over the TT2 1× plane must equal 1, then it follows that: c =0.5 We can then calculate Pr 2(T <T ) as follows: 12 12 1 2 (T1 <T2 )=. . 0.5 dt2 dt1 =.(0.5 t2 Pr 2 dt t )1 Page 3

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Sheet1 2 t =0 t =2tt =0 1 1 21 1 1 1 2 Err:509 tdt t 0 2 T1 t =0 t =0 0.5 If you cannot see why c =0.5 , then you will need to carry out one of the following integrations: 22 2 t 1 1 2 . .cdt 2 dt 1 =1 or . .cdt 1 dt 2 =1 121 21 t =0 t =tt =0 t =0 To calculate the probability you could have carried out the following integral: 1 . . . . 2t 2 2 0.5 dt1 dt 2 21 Page 4
Sheet1 t =0 t =0 Solution 8.5 Let X be the time until next claim for a Basic Policy, and let Y be the time until next claim for a Deluxe Policy. Since the two random variables are independent, the joint density is equal to the product of the two marginal densities. Hence: e-x /2 e-y /3 f( ) Err:509 × forx , y >0 xy , , Yy XY 23 Thus, the probability that the next claim is from a Deluxe Policy is: -x /2 -y /3 8 x8 xe e (Y <X)=. .f , x , . . × Pr () Err:520 dydx XY y dydx x=0 y=0 x=0 y=0 23 -x /2 -x /2 Page 5

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Sheet1 x .8e 8e .-y /3 Err:520 (-x /3 ) Err:520
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p chapter 8 solutions 3rd ed. fall 2007 - Probability Third...

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