1. In a large population, 35% of voters intend to vote for party A at the next election. A

random sample of 200 voters is selected from this population and asked which party

they will vote for.

Calculate, approximately, the probability that 80 or more of the people in this sample

intend to vote for party A. [4]

2.Let N be the random variable that describes the number of claims that an insurer

receives per month for one of its claim portfolios. We assume that N has a Poisson

distribution with E[N] = 50. The amount Xi

of each claim in the portfolio is normally

distributed with mean = 1,000 and variance 2 = 2002. The total amount of all

claims received during one month is

1

N

i

i

S X

=

=

with S = 0 for N = 0. We assume that N, X1, X2, ... are all independent of each other.

(i) Specify the type of the distribution of S. [1]

(ii) Calculate the mean and standard deviation of S. [3]

[Total 4] CT3 A20113 PLEASE TURN OVER

3.Let X1, X2, X3, X4, and X5 be independent random variables, such that Xi

~ gamma

with parameters i and for i = 1, 2, 3, 4, 5. Let

5

1

2 i

i

S X

=

= .

(i) Derive the mean and variance of S using standard results for the mean and

variance of linear combinations of random variables. [3]

(ii) Show that S has a chi-square distribution using moment generating functions

and state the degrees of freedom of this distribution. [4]

(iii) Verify the values found in part (i) using the results of part (ii). . [1]

[Total 8]

4.Consider two random variables X and Y, for which the variances satisfy V[X] = 5V[Y]

and the covariance Cov[X,Y] satisfies Cov[X,Y] = V[Y].

Let S = X + Y and D = X Y.

(i) Show that the covariance between S and D satisfies Cov[S,D] = 4V[Y]. [3]

(ii) Calculate the correlation coefficient between S and D. [3]

[Total 6] CT3 A20114

5.An insurance company distinguishes between three types of fraudulent claims:

Type 1: legitimate claims that are slightly exaggerated

Type 2: legitimate claims that are strongly exaggerated

Type 3: false claims

Every fraudulent claim is characterised as exactly one of the three types. Assume that

the probability of a newly submitted claim being a fraudulent claim of type 1 is 0.1.

For type 2 this probability is 0.02, and for type 3 it is 0.003.

(i) Calculate the probability that a newly submitted claim is not fraudulent. [1]

The insurer uses a statistical software package to identify suspicious claims. If a

claim is fraudulent of type 1, it is identified as suspicious by the software with

probability 0.5. For a type 2 claim this probability is 0.7, and for type 3 it is 0.9.

Of all newly submitted claims, 20% are identified by the software as suspicious.

(ii) Calculate the probability that a claim that has been identified by the software

as suspicious is:

(a) a fraudulent claim of type 1,

(b) a fraudulent claim of any type.

[5]

(iii) Calculate the probability that a claim which has NOT been identified as

suspicious by the software is in fact fraudulent