Ama Ama Ama Ama Assignment1

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AMA362 Further Statistical Methods Assignment 1 Due: Tuesday, 4 Oct. 2011 1. A utility function is given by   e−(w−100)2 /200 w < 100 u(w) =  2 − e−(w−100)2 /200 w ≥ 100. (a) Is u (w) ≥ 0? (b) For what range of w is u (w) < 0? [5 marks] [5 marks]

2. The decision maker has a utility function u(w) = −e−αw and is faced with a random loss X that has a chi-square distribution with n degrees of freedom. Suppose that 0 < α < 1/2. (a) Use u(w − G) = E[u(w − X)] to obtain an expression for G, the maximum insurance premium the decision maker will pay. (b) Prove that G > n = µ = E(X). [10 marks] [10 marks]

2 3. The loss random variable X has a p.d.f. given by f (x) = (a) Calculate E(X) and Var(X). (b) Consider a proportional policy where I(x) = kx 0 < k < 1, and a stop-loss policy where   0 x Var[X − Id (X)]. [10 marks] 4. Obtain the mean and variance of the claim random variable X where q = 0.05 and the claim amount random variable B is uniformly distributed between 0 and 20. [20 marks] 5. Let X and Y be independent and identically distributed with the uniform distribution on (0, 1). Let S = X + Y . (i) Determine the distribution function of S. (ii) Calculate E(S) and Var(S). [10 marks] [10 marks] 1 , 100 0 < x < 100. [10 marks]

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