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1、Course 16Form 03A 1.A survey of a groups viewing habits over the last year revealed the following information: (i)28% watched gymnastics (ii)29% watched baseball (iii)19% watched soccer (iv)14% watched gymnastics and baseball (v)12% watched baseball and soccer (vi)10% watched gymnastics and soccer (
2、vii) 8% watched all three sports. Calculate the percentage of the group that watched none of the three sports during the last year. (A)24 (B) 36 (C) 41 (D) 52 (E) 60 May 2003 - Course 1 May 20037Course 1 2.Each of the graphs below contains two curves. Identify the graph containing a curve representi
3、ng a function ( )yf x= and a curve representing its second derivative ( )yfx= . (A)(B) (C)(D) (E) Course 18Form 03A 3.Let f and g be differentiable functions such that ( ) ( ) 0 0 lim lim x x f xc g xd = = where cd. Determine ( )( ) ( )( ) 0 lim x cf xdg x f xg x . (A)0 (B) ( )( ) ( )( ) 00 00 cfdg
4、fg (C)( )( )00fg (D)cd (E)cd+ May 20039Course 1 4.The time to failure of a component in an electronic device has an exponential distribution with a median of four hours. Calculate the probability that the component will work without failing for at least five hours. (A)0.07 (B)0.29 (C)0.38 (D)0.42 (E
5、)0.57 Course 110Form 03A 5.An insurance company examines its pool of auto insurance customers and gathers the following information: (i)All customers insure at least one car. (ii)70% of the customers insure more than one car. (iii)20% of the customers insure a sports car. (iv)Of those customers who
6、insure more than one car, 15% insure a sports car. Calculate the probability that a randomly selected customer insures exactly one car and that car is not a sports car. (A)0.13 (B)0.21 (C)0.24 (D)0.25 (E)0.30 May 200311Course 1 6.Let X and Y be continuous random variables with joint density function
7、 8 for 01, 2 ( , )3 0otherwise. xyxxyx f x y = Calculate the covariance of X and Y. (A)0.04 (B)0.25 (C)0.67 (D)0.80 (E)1.24 Course 112Form 03A 7.( )( ) 24 02 Given 3 and 5, f x dxf x dx= () 2 0 calculate 2.fx dx (A)3 2 (B) 3 (C) 4 (D) 6 (E) 8 May 200313Course 1 8.An auto insurance company insures dr
8、ivers of all ages. An actuary compiled the following statistics on the companys insured drivers: Age of Driver Probability of Accident Portion of Companys Insured Drivers 16-20 21-30 31-65 66-99 0.06 0.03 0.02 0.04 0.08 0.15 0.49 0.28 A randomly selected driver that the company insures has an accide
9、nt. Calculate the probability that the driver was age 16-20. (A) 0.13 (B) 0.16 (C) 0.19 (D) 0.23 (E) 0.40 Course 114Form 03A 9.An insurance company determines it cannot write medical malpractice insurance profitably and stops selling the coverage. In spite of this action, the company will have to pa
10、y claims for many years on existing medical malpractice policies. The company pays 60 for medical malpractice claims the year after it stops selling the coverage. Each subsequent years payments are 20% less than those of the previous year. Calculate the total medical malpractice payments that the co
11、mpany pays in all years after it stops selling the coverage. (A) 75 (B)150 (C)240 (D)300 (E)360 May 200315Course 1 10.Let X and Y be continuous random variables with joint density function () 2 15 for , 0 otherwise. yxyx f x y = Let g be the marginal density function of Y. Which of the following rep
12、resents g? (A)( ) 15 for 01 0 otherwise yy g y = Calculate the difference between the 30th and 70th percentiles of X . (A) 35 (B) 93 (C)124 (D)231 (E)298 Course 128Form 03A 23.The time, T, that a manufacturing system is out of operation has cumulative distribution function 2 2 1for 2 ( ) 0otherwise.
13、 t F t t = The resulting cost to the company is 2 YT=. Determine the density function of Y, for 4y. (A) 2 4 y (B) 3/2 8 y (C) 3 8 y (D) 16 y (E) 5 1024 y May 200329Course 1 24.Let X represent the age of an insured automobile involved in an accident. Let Y represent the length of time the owner has i
14、nsured the automobile at the time of the accident. X and Y have joint probability density function () 2 1 10 for 210 and 01 64( , ) otherwise. 0 xy xy f x y = Calculate the expected age of an insured automobile involved in an accident. (A)4.9 (B)5.2 (C)5.8 (D)6.0 (E)6.4 Course 130Form 03A 25.An insu
15、rance policy pays for a random loss X subject to a deductible of C, where 01C . The loss amount is modeled as a continuous random variable with density function ( ) 2 for 01 0 otherwise. xx f x = Given a random loss X, the probability that the insurance payment is less than 0.5 is equal to 0.64 . Ca
16、lculate C. (A)0.1 (B)0.3 (C)0.4 (D)0.6 (E)0.8 May 200331Course 1 26.Let ( ) 2 4 28 x g x xx + = + . Determine all values of x at which g is discontinuous, and for each of these values of x, define g in such a manner so as to remove the discontinuity, if possible. (A)g is discontinuous only at 4 and 2. Define () 1 4 6 g= to make g continuous at 4. ( )2g cannot be defined to ma
