November 2001 - Course 1 SOA Solutions

More documents

Recommendations

Info

37. A The joint density of T1 and T 2 is given by −t −t f t , t = e e , t > 0 , t > 0 ( ) 1 2 1 2 1 2 Therefore, Pr X ≤ x = Pr 2T + T ≤ x [ ] [ ] 1 2 ⎡ = ∫∫ = − ⎢⎣ 1 1 x ( x−t2 ) x ( x−t2 ) 2 −t1 −t2 −t2 −t1 2 e e dt1dt2 e e dt2 0 0 ∫ ⎢ ⎥ 0 0 ⎡ ⎤ ⎛ ⎞ = ∫ − = − ⎣ ⎦ ⎝ ⎠ 1 1 1 1 x − x+ t2 x − x − t −t 2 2 2 2 −t2 2 2 e ⎢1 e ⎥dt2 ⎜e e e ⎟dt2 0 ∫0 ⎡ ⎤ = ⎢− e + e e ⎥ =− e + e e + − e ⎣ ⎦ 1 1 1 1 1 − x − t2 − x − x − x −t2 2 2 x − x 2 2 2 2 0 2 1 2 1 1 − x − x − x −x 2 2 −x = 1− e + 2e − 2e = 1− 2 e + e , x> 0 It follows that the density of X is given by 1 d ⎡ − x 2 − x ⎤ g ( x) = ⎢1− 2e + e ⎥ dx ⎣ ⎦ 1 − x 2 − x = e − e , x> 0 ⎤ ⎥⎦ 38. E X , X , and X denote annual loss due to storm, fire, and theft, respectively. In Let 1 2 3 addition, let Y Max( X , X , X ) Then = . 1 2 3 [ Y > ] = − [ Y ≤ ] = − [ X ≤ ] [ X ≤ ] [ X ≤ ] Pr 3 1 Pr 3 1 Pr 3 Pr 3 Pr 3 ( − e )( −3 e )( −3 e ) ( − e 3 )( − e )( −5 e ) = 1− 1− 1− 1− = 1− 1− 1− 1− 1 2 3 = 0.414 * Uses that if X has an exponential distribution with mean µ 1 Pr ( X ≤ x) = 1−Pr ( X ≥ x) = 1− ∫ e dt = 1−( − e ) x = 1−e µ ∞ x * −t µ −t µ ∞ −x µ Course 1 Solutions 23 November 2001
39. D We are given that P x =− x + 50x−25 ( ) 2 before the translation occurs. The revised profit function P ( x) follows: P* x = P x−2 −3 ( ) ( ) 2 ( x ) ( x ) =− − 2 + 50 −2 −25−3 2 =− x + x− + x− − 2 =− x + x− 4 4 50 100 28 54 132 * may be determined as 40. B Observe that E X + Y = E X + E Y = 50 + 20 = 70 [ ] [ ] [ ] [ ] [ ] [ ] [ ] Var X + Y = Var X + Var Y + 2 Cov X , Y = 50 + 30 + 20 = 100 for a randomly selected person. It then follows from the Central Limit Theorem that T is approximately normal with mean ET = 100 70 = 7000 [ ] ( ) and variance Var T = 100 100 = 100 [ ] ( ) 2 Therefore, ⎡T −7000 7100 −7000⎤ Pr[ T < 7100] = Pr ⎢ < ⎣ 100 100 ⎥ ⎦ = Pr[ Z < 1] = 0.8413 where Z is a standard normal random variable. Course 1 Solutions 24 November 2001
Page 1 and 2: Course 1 Solutions November 2001 Ex
Page 3 and 4: 3. A Since 3 4 2 5 S = 175L A , we
Page 5 and 6: 7. A Cov C , C = Cov X + Y, X + 1.2
Page 7 and 8: 8. D The function F() t , 1≤t ≤
Page 9 and 10: 12. E Observe that x f′ x = f′
Page 11 and 12: 17. B Note that 20 [ > ] = ( − )
Page 13 and 14: 21. D Let N1 and N 2 denote the num
Page 15 and 16: 24. D 1 1 4 2 100 x y must be maxim
Page 17 and 18: 27. D Let X denote the number of em
Page 19 and 20: 29. B Let X and Y denote repair cos
Page 21 and 22: 31. E The general solution of the d
Page 23: 35. E If X is the random variable r

November 2001 - Course 1 SOA Solutions

Create successful ePaper yourself

Delete template?

Save as template?