# chapter6 - 1 Chapter 6 61 fX(x = 1 4 0

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1 Chapter 6 6–1. f X ( x ) = 1 4 ; 0 < x < 4 P ± 1 2 < X < 7 4 = Z 7 / 4 1 / 2 dx 4 = 5 16 , P ± 9 4 < X < 27 8 = Z 27 / 8 9 / 4 dx 4 = 9 32 6–2. f X ( x ) = 4 34 ; 143 4 < x < 177 4 P ( X < 40) = Z 40 143 / 4 4 34 dx = 1 2 P (40 < X < 42) = Z 42 40 4 34 dx = 4 17 6–3. f X ( x ) = 1 2 , 0 x 2 F Y ( y ) = P ( Y y ) = P ± X y - 5 2 = Z y - 5 2 0 dx 2 = y - 5 4 So f Y ( y ) = 1 4 , 5 < y < 9 6–4. The p.d.f. of proﬁt, X , is f X ( x ) = 1 2000 ; 0 < x < 2000 Y = Brokers Fees = 50 + 0 . 06 X F Y ( y ) = P (50 + 0 . 06 X y ) = P ± X y - 50 0 . 06 = Z y - 50 0 . 06 0 dx 2000 = y - 50 120 f Y ( y ) = 1 120 ; 50 < y < 170

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2 6–5. M X ( t ) = E ( e tX ) = Z β α e tx dx β - α = e tx ( β - α ) t β α = e - e ( β - α ) t Using L’Hˆospital’s rule when necessary, we obtain E ( X ) = M 0 X (0) = 1 β - α £ t - 1 βe - t - 2 e - t - 1 αe + t - 2 e / t =0 = 1 β - α £ β 2 e - β 2 e / 2 - α 2 e + α 2 e / 2 / t =0 = β + α 2 and E ( X 2 ) = M 00 X (0) = 1 β - α £ t - 1 β 2 e - βe t - 2 + 2 e t - 3 - t - 2 βe - t - 1 α 2 e + αe t - 2 + t - 2 αe - 2 e t - 3 / t =0 = 1 β - α β 3 - α 3 3 V ( X ) = E ( X 2 ) - [ E ( X )] 2 = ( β - α ) 2 12 6–6. E ( X ) = β + α 2 = 0 β + α = 0 V ( X ) = ( β - α ) 2 12 = 1 β 2 - 2 αβ + α 2 = 12 α = - 3 , β = + 3
3 6–7. The CDF for Y is y F Y ( y ) y < 1 0 1 y < 2 0.3 2 y < 3 0.5 3 y < 4 0.9 y > 4 1 Generate realizations of u i Uniform[0,1] random numbers as described in Section 6–6; use these in the inverse as y i = F - 1 Y ( u i ), i = 1 , 2 ,... . For example, if u 1 = 0 . 623, then y 1 = F - 1 Y (0 . 623) = 3. 6–8. f X ( x ) = 1 4 ; 0 < x < 4 = 0; otherwise The roots of y 2 + 4 xy + ( x + 1) = 0 are real if 16 x 2 - 4( x + 1) 0, or if ( x - 1 8 ) 2 - 17 64 0 or where x 1 8 (1 - 17) or x 1 8 (1 + 17) P ± X 1 8 (1 - 17) = 0 P ± X 1 8 (1 + 17) = Z 4 1 8 (1+ 17) dx 4 = 1 32 (31 - 17) 6–9. M X ( t ) = Z 0 e tx λe - λx dx = λ Z 0 e x ( t - λ ) dx, which converges if t < λ . Thus for t < λ , M X ( t ) = λ t - λ [ e x ( t - λ )] x =0 = λ λ - t = 1 1 - λ/t , t < λ

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4 E ( X ) = M 0 X (0) = £ λ ( λ - t ) - 2 / t =0
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chapter6 - 1 Chapter 6 61 fX(x = 1 4 0

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