北美精算师真题course1

May 2003 - Course 1

1.A survey of a group’s viewing habits over the last year revealed the followinginformation:

(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(vii) 8% watched all three sports.

Calculate the percentage of the group that watched none of the three sportsduring the last year.

(A)24(B) 36(C) 41(D) 52(E)

60

6Form 03A

Course 1

2.Each of the graphs below contains two curves.

Identify the graph containing a curve representing a function y=f(x) and a curverepresenting its second derivative y=f′′(x) .

May 2003(A)

(C)

(E)

(B)

(D)

7Course 1

3.Let f and g be differentiable functions such that

where c≠d.

Determine lim

cf(x) dg(x)

x→0fx gx .(A)0

(B)cf′(0) dg′(0)f′0 g′0(C)f′(0) g′(0)(D)c d(E)

c+d

limx→0f(x)=c

limx→0

g(x)=d

8Form 03A

Course 1

4.The time to failure of a component in an electronic device has an exponentialdistribution with a median of four hours.

Calculate the probability that the component will work without failing for at leastMay 2003five hours.

(A)0.07(B)0.29(C)0.38(D)0.42(E)

0.57

9Course 1

5.An insurance company examines its pool of auto insurance customers and gathers thefollowing 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 insure more than one car, 15% insure a sports car.

Calculate the probability that a randomly selected customer insures exactly one car andthat car is not a sports car.

(A)0.13(B)0.21(C)0.24(D)0.25(E)

0.30

10Form 03A

Course 1

6.Let X and Y be continuous random variables with joint density function

May 2003 f(x,y)= 8 3

xy

0

Calculate the covariance of X and Y.

(A)0.04(B)0.25(C)0.67(D)0.80(E)

1.24

for 0≤x≤1, x≤y≤2xotherwise.

11Course 1

7.

Given ∫f(x)dx=3 and ∫f(x)dx=5,

24

2

calculate ∫2

f(2x)dx.

(A)2

(B) 3(C) 4(D) 6(E)

8

Form 03A

Course 112

8.May 2003An auto insurance company insures drivers of all ages. An actuary compiled thefollowing statistics on the company’s insured drivers:

Age ofProbability Portion of Company’sDriverof Accident Insured Drivers

16-200.060.0821-300.030.1531-650.020.4966-990.040.28

A randomly selected driver that the company insures has an accident.

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

13Course 1

9.An insurance company determines it cannot write medical malpractice insuranceprofitably and stops selling the coverage. In spite of this action, the company willhave to pay claims for many years on existing medical malpractice policies.

The company pays 60 for medical malpractice claims the year after it stops selling thecoverage. Each subsequent year’s payments are 20% less than those of the previous year.

Calculate the total medical malpractice payments that the company pays in all years afterit stops selling the coverage.

(A) 75(B)150(C)240(D)300(E)

360

14Form 03A

Course 1

10.May 2003Let X and Y be continuous random variables with joint density function

f(x,y)=

15y for x2≤y≤x

0 otherwise.

Let g be the marginal density function of Y.

Which of the following represents g?

(A)

g(y)=

15y for 0<y<1

0 otherwise 15y2

(B)

g(y)=

for x2<y<x 2

0 otherwise 15y2

(C)

g(y)=

for 0<y<1 2

0 otherwise3/21/(D)

g(y)=

15y(1 y2) for x2

<y<x

0 otherwise(E)

g(y)=

15y3/2(1 y1/2) for 0<y<1

0 otherwise

15Course 1

11.The value of a particular investment changes over time according to the function

S(t)0.1 0.25t =5000e

e

,

where S(t) is the value after t years.

Calculate the rate at which the value of the investment is changing after 8 years.

(A) 618(B) 1,934(C) 2,011(D) 7,735(E)

10,468

16Form 03A

Course 1

12.Let X be a continuous random variable with density function

x

for 2≤x≤4f(x)= 10

0 otherwise.

May 2003Calculate the expected value of X.

(A)

15(B)

35

(C) 1(D)

2815(E)

125

17Course 1

13.A charity receives 2025 contributions. Contributions are assumed to be independentand identically distributed with mean 3125 and standard deviation 250.

Calculate the approximate 90th percentile for the distribution of the total contributionsreceived.

(A)6,328,000(B)6,338,000(C)6,343,000(D)6,784,000(E)

6,977,000

Form 03A

Course 118

14.Let f be a differentiable function such that:

f(x+h) f(x)=3x2h+3xh2+h3+2h for all x and hMay 2003f(0)=1

Let g(x)=e xf(x).

Calculate g′(3).

(A) 34e 3(B) 29e 3(C) 5e 3(D) 4e 3(E)

63e 3

19Course 1

15.An insurance policy pays a total medical benefit consisting of two parts for each claim.Let X represent the part of the benefit that is paid to the surgeon, and let Y represent thepart that is paid to the hospital. The variance of X is 5000, the variance of Y is 10,000,and the variance of the total benefit, X+Y, is 17,000.

Due to increasing medical costs, the company that issues the policy decides to increase

X by a flat amount of 100 per claim and to increase Y by 10% …… 此处隐藏：3460字，全部文档内容请下载后查看。喜欢就下载吧 ……