complete all using simple calculations

1 Consider the following two random samples of ten observations which come from the

distributions of random variables which assume non-negative integer values only.

Sample 1: 7 4 6 11 5 9 8 3 5 5

sample mean = 6.3, sample variance = 6.01

Sample 2: 8 3 5 11 2 4 6 12 3 9

sample mean = 6.3, sample variance = 12.46

One sample comes from a Poisson distribution, the other does not.

State, with brief reasons, which sample you think is likely to be which. [2]

2 A random sample of 200 policy surrender values (in units of 1,000) yields a mean of

43.6 and a standard deviation of 82.2.

Determine a 99% confidence interval for the true underlying mean surrender value for

such policies. [3]

3 It is assumed that claims on a certain type of policy arise as a Poisson process with

claim rate per year.

For a group of 150 independent policies of this type, the total number of claims during

the last calendar year was recorded as 123.

Determine an approximate 95% confidence interval for the true underlying annual

claim rate for such a policy. [4]

4 The sample correlation coefficient for the set of data consisting of the three pairs of

values

(1,2) , (0,0) , (1,1)

is 0.982. After the x and y values have been transformed by particular linear functions,

the data become:

(2,2) , (6,4) , (10,7).

State (or calculate) the correlation coefficient for the transformed data. [2] CT3 A20073 PLEASE TURN OVER

5 The number of claims arising in one year from a group of policies follows a Poisson

distribution with mean 12. The claim sizes independently follow an exponential

distribution with mean 80 and they are independent of the number of claims.

The current financial year has six months remaining.

Calculate the mean and the standard deviation of the total claim amount which arises

during this remaining six months. [4]

6 Consider the discrete random variable X with probability function

1

4 ( ) , 0,1, 2, ...

5x fx x + = =

(i) Show that the moment generating function of the distribution of X is given by

1 ( ) 4(5 )t Mt e X = ,

for e

t < 5. [3]

(ii) Determine E[X] using the moment generating function given in part (i). [3]

[Total 6]

7 A charity issues a large number of certificates each costing 10 and each being

repayable one year after issue. Of these certificates, 1% are randomly selected to

receive a prize of 10 such that they are repaid as 20. The remaining 99% are repaid

at their face value of 10.

(i) Show that the mean and standard deviation of the sum repaid for a single

purchased certificate are 10.1 and 0.995 respectively. [2]

Consider a person who purchases 200 of these certificates.

(ii) Calculate approximately the probability that this person is repaid more than

2,040 by using the Central Limit Theorem applied to the total sum repaid.

[3]

(iii) An alternative approach to approximating the probability in (ii) above is based

on the number of prize certificates the person is found to hold. This number

will follow a binomial distribution.

Use a Poisson approximation to this binomial distribution to approximate the

probability that this person is repaid more than 2,040. [3]

(iv) Comment briefly on the comparison of the two approximations above given

that the exact probability using the binomial distribution is 0.0517. [1]

[Total 9] CT3 A20074

8 A random sample of size n is taken from a distribution with probability density

function

1 () , 0 (1 ) fx x

x +

= < <

+

where is a parameter such that > 0.

(i) Show by evaluating the appropriate integral that, in the case > 1, the mean

of this distribution is given by 1

1

.

[Hint: when integrating, write x = (1 + x) - 1 and exploit the fact that the

integral of a density function is unity over its full range.] [3]

(ii) Determine the method of moments estimator of . [2]

[Total 5]

9 Consider three random variables X, Y, and Z with the same variance 2 = 4. Suppose

that X is independent of both Y and Z, but Y and Z are correlated, with correlation

coefficient YZ = 0.5.

(i) Calculate the covariance between X and U, where U = Y+Z. [1]

(ii) Calculate the covariance between Z and V, where V = 3X - 2Y. [2]

(iii) Calculate the variance of W, where W = 3X - 2Y + Z. [2]

[Total 5]

10 A random sample of insurance policies of a certain type was examined for each of

four insurance companies and the sums insured (yij, for companies i = 1, 2, 3, 4)

under each policy are given in the table below (in units of 100):

Company Total

1 58.2 57.2 58.4 55.8 54.9 284.5

2 56.3 54.5 57.0 55.3 223.1

3 50.1 54.2 55.5 159.8

4 52.9 49.9 50.0 51.7 204.5

For these data, 871.9 ij i j y = and 2 47,633.53 ij i j y =CT3 A20075 PLEASE TURN OVER

Consider the ANOVA model , 1,..., 4, 1,..., Y ei j n ij i ij i = + + = = , where Yij is the jth

sum insured for company i, ni

is the number of responses for company i,

2 ~ (0, ) ij e N are independent errors, and 4

1 0 i i i

n = = .

(i) Calculate estimates of the parameters and , 1, 2, 3, 4 i i = . [3]

(ii) Test the hypothesis that there are no differences in the means of the sums

insured under such policies by the four companies. [5]

[Total 8]

11 The number of claims, X, which arise in a year on each policy of a particular class is

to be modelled as a Poisson random variable with mean . Let X = (X1, X2, ..., Xn) be

a random sample of size n from the distribution of X, and let

1

1 n

i

i

X X

n =

= .

Suppose that it is required to estimate , the mean number of claims on a policy.

(i) Show that

, the maximum likelihood estimator of , is given by

= X . [3]

(ii) Derive the Cramer-Rao lower bound (CRlb) for the variance of unbiased

estimators of . [4]

(iii) (a) Show that

is unbiased for and that it attains the CRlb.

(b) Explain clearly why, in the case that n is large, the distribution of

can

be approximated by

~ , N

n

.

[3]

(iv) (a) Show that, in the case n = 100, an approximate 95% confidence

interval for is given by

x 0.196 x .

(b) Evaluate the confidence interval in (iv)(a) based on a sample with the

following composition:

observation 0 1 2 3 4 5 6 7

frequency 11 28 19 28 9 2 2 1