北美精算师 exam P 2003 真题

c f ( x ) − dg ( x ) . x →0 f ( x ) − g ( x )
(A) (B) (C) (D) (E)
0 cf ′ ( 0 ) − dg ′ ( 0 ) f ′ ( 0) − g′ ( 0)
f ′ ( 0) − g′ ( 0) c−d c+d
Course 1
8
Form 03A
Calculate the covariance of X and Y.
(A) (B) (C) (D) (E)
0.04 0.25 0.67 0.80 1.24
May 2003
11
Course 1
7.
Given
∫
2 0
f ( x ) dx = 3 and
∫ f ( x ) dx = 5,
2
4
calculate
(A) (B) (C) (D) (E)
0.07 0.29 0.38 0.42 0.57
May 2003
9
Course 1
5.
An insurance company examines its pool of auto insurance customers and gathers the following information:
15
Course 1
11.
The value of a particular investment changes over time according to the function
S ( t ) = 5000e
0.25t 0.1 e

,
where S ( t ) is the value after t years.
Calculate the probability that a randomly selected customer insures exactly one car and that car is not a sports car.
(A) (B) (C) (D) (E)
0.13 0.21 0.24 0.25 0.30
∫ f ( 2 x ) dx .
0
2
(A) (B) (C) (D) (E)
32
3 4 6 8
Course 1
12
Form 03A
8.
An auto insurance company insures drivers of all ages. An actuary compiled the following statistics on the company’s insured drivers:
Course 1
18
Form 03A
14.
Let f be a differentiable function such that:
f ( x + h ) − f ( x ) = 3 x 2 h + 3 xh 2 + h3 + 2h for all x and h f (0) = 1
Let g ( x ) = e − x f ( x ) .
Calculate the total medical malpractice payments that the company pays in all years after it stops selling the coverage.
(A) (B) (C) (D) (E)
75 150 240 300 360
Age of Driver 16-20 21-30 31-65 66-99
Probability of Accident 0.06 0.03 0.02 0.04
Portion of Company’s Insured Drivers 0.08 0.15 0.49 0.28
A randomly selected driver that the company insures has an accident.
The company pays 60 for medical malpractice claims the year after it stops selling the coverage. Each subsequent year’s payments are 20% less than those of the previous year.
Course 1
10
Form 03A
6.
Let X and Y be continuous random variables with joint density function
8 xy f ( x, y ) = 3 0
for 0 ≤ x ≤ 1, x ≤ y ≤ 2 x otherwise.
3/ 2 1/ 2 2 15 y (1 − y ) for x < y < x g ( y) = otherwise 0 3/ 2 1/ 2 15 y (1 − y ) for 0 < y < 1 g ( y) = otherwise 0
(B)
(C)
(D)
(E)
May 2003
Calculate the approximate 90th percentile for the distribution of the total contributions received.
(A) (B) (C) (D) (E)
6,328,000 6,338,000 6,343,000 6,784,000 6,977,000
Each of the graphs below contains two curves.
Identify the graph containing a curve representing a function y = f ( x ) and a curve representing its second derivative y = f ′′ ( x ) .
Calculate the percentage of the group that watched none of the three sports during the last year.
(A) (B) (C) (D) (E)
24 36 41 52 60
Course 1
6
Form 03A
2.
Calculate the probability that the driver was age 16-20.
(A) (B) (C) (D) (E)
0.13 0.16 0.19 0.23 0.40
May 2003
13
Course 1
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 pay claims for many years on existing medical malpractice policies.
Calculate the rate at which the value of the investment is changing after 8 years.
(A) (B) (C) (D) (E)
618 1,934 2,011 7,735 10,468
Course 1
16
Form 03A
12.
(i) (ii) (iii) (iv)
All customers insure at least one car. 70% of the customers insure more than one car. 20% of the customers insure a sports car. Of those customers who insure more than one car, 15% insure a sports car.
(A)
(B)
(C)
(D)
(E)
May 2003
7
Course 1
3.
Let f and g be differentiable functions such that
lim f ( x ) = c lim g ( x ) = d
x →0 x →0
where c ≠ d .
Determine lim
May 2003 - Course 1
1.
A survey of a group’s viewing habits over the last year revealed the following information:
(i) (ii) (iii) (iv) (v) (vi) (vii)
28% watched gymnastics 29% watched baseball 19% watched soccer 14% watched gymnastics and baseball 12% watched baseball and soccer 10% watched gymnastics and soccer 8% watched all three sports.
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.
Let X be a continuous random variable with density function
x for − 2 ≤ x ≤ 4 f ( x ) = 10 0 otherwise.
Calculate the expected value of X.
(A)
1 5 3 5
(B) (C) (D)
1
28 15 12 5
(E)
May 2003
17
Course 1
13.
A charity receives 2025 contributions. Contributions are assumed to be independent and identically distributed with mean 3125 and standard deviation 250.
Course 1
14
Form 03A
10.
Let X and Y be continuous random variables with joint density function
15 y for x 2 ≤ y ≤ x f ( x, y ) = otherwise. 0
Let g be the marginal density function of Y.
Calculate g′(3).
(A)
−34e −3 −29e −3 −5e −3 −4e −3 63e −3
(B) (C) (D) (E)
May 2003
19
Course 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 the part 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.
Which of the following represents g?
(A)
15 y for 0 < y < 1 g ( y) = otherwise 0 15 y 2 for x 2 < y < x g ( y) = 2 0 otherwise 15 y 2 for 0 < y < 1 g ( y) = 2 0 otherwቤተ መጻሕፍቲ ባይዱse