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# Exam1Bans - Exam 1 1 Version B yellow 2 2 x y 81 0 Answers...

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Page 1 of 6 Exam 1 ( Version B – yellow ) ( Answers ) 1. Consider two continuous random variables X and Y with joint p.d.f. f X, Y ( x , y ) = ± ² ³ < < < < otherwise 0 0 , 0 81 2 2 K K y x y x a) (5) Find the value of K so that f X, Y ( x , y ) is a valid joint p.d.f. 1 = ´ ´ K K dy dx y x 0 0 2 81 2 = 243 5 K . µ K = 3 . b) (5) Find P ( X > 3 Y ). P ( X > 3 Y ) = ´ ´ · ¸ ¹ ¹ ¹ º » 3 0 3 0 2 81 2 dx dy y x x = ´ 3 0 4 729 1 dx x = 15 1 . OR P ( X > 3 Y ) = ´ ´ · ¸ ¹ ¹ ¹ º » 1 0 3 3 2 81 2 dy dx y x y = … = 15 1 .

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Page 2 of 6 2. (7) Determine the mean and variance of a random variable with the following moment-generating function: M X ( t ) = e t ( 7 + 8 t ) = exp [ t ( 7 + 8 t ) ]. M X ( t ) = exp [ t ( 7 + 8 t ) ] = exp ( 7 t + 8 t 2 ). Normal distribution: M X ( t ) = 2 2 2 ± t t e + . X has a Normal distribution with μ = 7 and σ 2 / 2 = 8. E ( X ) = μ = 7 , Var ( X ) = σ 2 = 16 . OR M X ' ( t ) = ( 7 + 16 t ) exp ( 7 t + 8 t 2 ). E ( X ) = M X ' ( 0 ) = 7 . M X " ( t ) = 16 exp ( 7 t + 8 t 2 ) + ( 7 + 16 t ) 2 exp ( 7 t + 8 t 2 ). E ( X 2 ) = M X " ( 0 ) = 65. Var ( X ) = 65 – 7 2 = 16 . 3. (6) A machine fastens plastic screw-on caps onto containers of motor oil. If the machine applies more torque than the cap can withstand, the cap will break. Both the torque
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Exam1Bans - Exam 1 1 Version B yellow 2 2 x y 81 0 Answers...

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