STAT3010/6075 - Statistical Methods in Insurance
Statistical Methods in Insurance
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: THEend8_
STAT3010/6075 - Statistical Methods in Insurance
Mock Class Test
Answer all 15 questions.
A correct answer scores +1, an incorrect answer and no answer scores 0. Some
questions may take longer to answer than others.
This test contributes 10% to the overall module assessment.
Time allowed: 45 minutes.
2
1. Moral hazard in connection with an insurance policy refers to
A. the temptation for the insured to make the insured event more likely to happen
B. the need to avoid selling insurance policies to those with a criminal record
C. the fact that it is illegal to insure against risks connected with criminal
activities
D. the absence of an insurable interest in the risks covered by the insurance
policy
2. A random variable X has a moment generating function . Evaluation of
gives
A. E[X2]
B. E[(log X)2 ]
C. Var(log X)
D. Var(X)
3. If X and Y are jointly distributed random variables, then E[X] can be expressed as
A. YXEE XY |
B. YXEE YX |
C. XYEE XY |
D. XYEE YX |
3
4. If a random variable X has a moment generating function then E[Xa], where a
is a constant, is
A.
B. )(log aM X
C.
D.
5. The value of
dx
x
c
+
0
4
is
A.
3
2c
B. λ3
C. c
D. c
2
6. The value of
3
n6
1n+
0=n
is
A. 6
B. 2
C. 3/2
D. 3
4
7. The collective risk model
NXXXS +++= 21
assumes that
I Xi are independent
II N is a Poisson variable with parameter λ
III S is independent of N
IV the Xi have finite variance
V Xi and N are independent
How many of the above statements are true?
A. two
B. three
C. four
D. five
8. In an insurance portfolio described by a collective risk model, the term period of the
portfolio is
A. one year
B. a random period of time
C. any predetermined time period
D. one year or six months
9. For the collective risk model, which of the following is true
A. P(S = 0) = P(N = 0)
B. P(S = 0) = P(N = 1)
C. P(S = 0) is undefined
D. P(S = 0) = 0
5
10. The tail area probabilities of four different loss distributions are
I
xe 1.0−
II
xe 03.0−
III
xe 03.0−
IV
203.0 xe−
The distribution with the fattest tail is
A. I
B. II
C. III
D. IV
11. The random variable S in a collective risk model has mean 20 and variance 64.
If S is modelled using a normal distribution, then P[S > 40] equals
A. 0.0062
B. 0.9938
C. 0.1587
D. 0.4880
6
12. Given =λ, the random variable X has an exponential distribution with parameter λ,
i.e.
λ > 0, x > 0.
The parameter itself varies and is a random variable which has an exponential
distribution with parameter , i.e.,
.0,0,)( = −
ef
The unconditional probability density function of X, fX(x), is
A.
xe )()( +−+
B.
+
x
C.
2)( +
x
D.
)(
1
−− xex
13. From the following distributions
I Gamma (2,5)
II Gamma (2,10)
III Gamma (5,2)
which is the more skewed?
A. I
B. II
C. III
D. both I & II
E. both I & III
7
14. The table below shows the frequency distribution of the number of claims recorded
on each of 400 motor insurance policies.
Number of claims, x Observed number of policies
0 264
1 83
2 31
3 13
4 7
5 2
The sample mean for these data is 0.555 and the sample standard deviation is
0.953545.
A negative binomial distribution for the number of claims is to be fitted to the data.
The method of moments estimate of the parameter p from the probability function
...,2,1,0,)1(
!)!1(
)!1(
)( =−
−
−+
= xpp
x
x
xP xX
is
A. 1.6383
B. 0.6104
C. 0.5820
D. 1.7181
8
15. For the data in Question 14, a Poisson distribution has been fitted for the number of
claims and the corresponding fitted number of policies using the Poisson distribution
is shown in the third column below.
Number Observed number Fitted number
of claims, x of policies of policies
0 264 230
1 83 127
2 31 35
3 13 7
4 7 1
5 2 0
Recall the sample mean and standard deviation for these data, given in Question 14.
Consider the following statements
I The maximum likelihood estimate of the parameter of the Poisson
distribution for these data is 0.555.
II The value of the chi-squared test statistic for these data is greater than 20.
III The Poisson distribution is a poor fit for the observed data.
IV Under the hypothesis that the Poisson fits the data well, the goodness of fit
test statistic has a
2 distribution with 5 degrees of freedom.
How many of the above statements are true?
A. one
B. two
C. three
D. four