1 The stem and leaf plot below gives the surrender values (to the nearest 1,000) of 40

endowment policies issued in France and recently purchased by a dealer in such

policies in Paris. The stem unit is 10,000 and the leaf unit is 1,000. 5 3

5 6

6 02

6 5779

7 122344

7 556677899

8 1123444

8 567778

9 024

9 6

Determine the median surrender value for this batch of policies. [2]

2 In a certain large population 45% of people have blood group A. A random sample of

300 individuals is chosen from this population.

Calculate an approximate value for the probability that more than 115 of the sample

have blood group A. [3]

3 A random sample of size 10 is taken from a normal distribution with mean = 20 and

variance 2

= 1.

Find the probability that the sample variance exceeds 1, that is find P(S

2

> 1). [3]

4 In a one-way analysis of variance, in which samples of 10 claim amounts () from

each of three different policy types are being compared, the following means were

calculated:

1 2 3

y 276.7 , y 254.6 , y 296.3

with residual sum of squares SSR

given by

3 10

2

1 1

( ) 15,508.6 R ij i

i j

SS y y

Calculate estimates for each of the parameters in the usual mathematical model, that

is, calculate 1 2 3

, , , , and 2

. [4]CT3 A2006 3 PLEASE TURN OVER

5 A large portfolio of policies is such that a proportion p (0 < p < 1) incurred claims

during the last calendar year. An investigator examines a randomly selected group of

25 policies from the portfolio.

(i) Use a Poisson approximation to the binomial distribution to calculate an

approximate value for the probability that there are at most 4 policies with

claims in the two cases where (a) p = 0.1 and (b) p = 0.2. [3]

(ii) Comment briefly on the above approximations, given that the exact values of

the probabilities in part (i), using the binomial distribution, are 0.9020 and

0.4207 respectively. [2]

[Total 5]

6 One variable of interest, T, in the description of a physical process can be modelled as

T = XY where X and Y are random variables such that X ~ N(200, 100) and Y depends

on X in such a way that Y|X = x ~ N(x, 1).

Simulate two observations of T, using the following pairs of random numbers

(observations of a uniform (0, 1) random variable), explaining your method and

calculations clearly:

Random numbers

0.5714 , 0.8238

0.3192 , 0.6844

[6]

7 Let (X1

, X2

, , Xn

) be a random sample from a uniform distribution on the interval

( , ), where is an unknown positive number.

A particular sample of size 5 gives values 0.87, 0.43, 0.12, 0.92, and 0.58.

(i) Draw a rough graph of the likelihood function L( ) against for this sample.

[3]

(ii) State the value of the maximum likelihood estimate of . [2]

[Total 5]CT3 A2006 4

8 The events that lead to potential claims on a policy arise as a Poisson process at a rate

of 0.8 per year. However the policy is limited such that only the first three claims in

any one year are paid.

(i) Determine the probabilities of 0, 1, 2 and 3 claims being paid in a particular

year. [2]

(ii) The amounts (in units of 100) for the claims paid follow a gamma

distribution with parameters = 2 and = 1.

Calculate the expectation of the sum of the amounts for the claims paid in a

particular year. [3]

(iii) Calculate the expectation of the sum of the amounts for the claims paid in a

particular year, given that there is at least one claim paid in the year. [2]

[Total 7]

9 The total claim amount on a portfolio, S, is modelled as having a compound

distribution

S = X1

+ X2 + + XN

where N is the number of claims and has a Poisson distribution with mean , Xi

is the

amount of the i

th claim, and the Xi s are independent and identically distributed and

independent of N. Let MX

(t) denote the moment generating function of Xi

. (i) Show, using a conditional expectation argument, that the cumulant generating

function of S, CS

(t), is given by

CS

(t) = MX

(t) 1}.

Note: You may quote the moment generating function of a Poisson random

variable from the book of Formulae and Tables. [4]

(ii) Calculate the variance of S in the case where = 20 and X has mean 20 and

variance 10. [2]

[Total 6]