An insurance policy covers insureds with two levels of risks { denoted by Level 1 and

Level 2. Each insured will have at most one claim. The probability to make a claim and

the severity of claims are given below.

{ Level 1 insureds have 40% males and 60% females. 20% of males and 25% of

females are expected to le a claim, and the size of each claim may be 5, 10 or 20

with probabilities 0.3, 0.5 and 0.2 respectively.

{ Level 2 insureds have 50% males and 50% females. 30% of males and 20% of females

are expected to le a claim, with each claim size equally likely to be 10 or 20.

{ Level 1 insureds are twice of Level 2 insureds.

In this case, the risk parameter takes four values, labeled as i; i = 1; 2; 3; 4:

1: male insured of Level 1; 2: female insured of Level 1;

3: male insured of Level 2; 4: female insured of Level 2.

(a) Derive the conditional probability mass function pXj(xj) = P(X = xj = ).

[2 marks]

(b) Derive the conditional probability mass function pX9jX(xjx) = P(X9 = xjX = x)

and the expectation for next positive claim E[X9jfX = xg [ fX9 > 0g], where X =

(X1;X2;X3;X4;X5;X6;X7;X8) and x = (10; 20; 10; 20; 10; 20; 10; 20). [8 marks]

(c) According to the experience of (b), determine the Buhlmann credibility premium.

3. (a) Claim sizes are uniform on [0, O + 20] for each policyholder, where O follows an exponential distribution with mean 10. A policyholder is selected at random and a claim of 30 is observed. Determine the expected value of the next claim from the same policyholder. [3 marks] (b) The number of claims on an insurance coverage follows a Poisson distribution. The mean of the Poisson distribution, A varies by insured, with the following probabili- ties: A Prior probability D.1 0.5 0.2 0.3 0.3 0.2 A randomly selected insured submits 1 claim in 5 years. Calculate the probability that this insured submits at least 1 claim in the sixth year. [2 marks] (c) You are given the following: Ten urns contain balls; For k = 1, 2, 3,4,5, in Urn k: 10%% of the balls are marked 0, 10% of the balls are marked 2, and the others are marked 0.5t; For k = 6, 7,8, 9, 10, in Urn k: 10% of the balls are marked 0 and the others are marked 0.1kt. An urn is randomly selected. A ball is then randomly selected from this urn, observed, and replaced. You wish to estimate the expected value of the number on the second ball randomly selected from this same urn. (1) Find the integer values of t so that Buhlmann credibility of the first observation is less than 0.15. [3 marks] (2) Determine the limit of Buhlmann credibility of the first observation as t goes to infinity. [2 marks]