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This is the Exam of Statistical Science which includes Fixed Point Theorem, Erlang Fixed Point, Loss Network, Fixed Routing, Carefully, Relating, Approximation, Mathematical, Wardrop Equilibrium etc. Key important points are: Distributed Claims, Insurance Policies, Identically Distributed Claims, Generating Function, Probability, Generating, Moment, Total Claim, Terms, Exponentially
Typology: Exams
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Friday 7 June 2002 9 to 11
Attempt THREE questions There are four questions in total
The questions carry equal weight
1 A portfolio of insurance policies gives rise to N claims in a particular time period, with independent, identically distributed claims X 1 , X 2 ,.. ., independent of N.
Find the moment generating function of the total claim S = X 1 +.. .+ XN in terms of the probability generating function GN of N and the moment generating function MX of X 1.
Suppose that
P(N = n) = pn =
k + n − 1 n
(1 − p)npk, n = 0, 1 , 2 ,...
where k is a positive integer and 0 < p < 1, and suppose that the claims are exponentially distributed with mean (^) λ^1. Show that S has the same distribution as the total claim
amount S˜ when the number of claims N˜ has a Binomial distribution. Show that this Binomial distribution has parameters k and 1 − p, and that the associated claim sizes are exponentially distributed.
Show that
P(S > x) =
n=
pn
n∑− 1
j=
e−λx(λx)j j!
, x > 0
and find a similar expression for P( S > x˜ ). Explain why the relationship between S and S˜ is of practical use in calculation of P(S > x).
2 In a classical risk model with relative safety loading ρ > 0, claims arrive in a Poisson process, rate λ > 0, and claims X 1 , X 2 ,... are independent exponential random variables with mean μ independent of the claim arrival process. Define the adjustment coefficient R, and find R in terms of ρ and μ.
Define the probability of ruin with initial capital u > 0, and state the Lundberg inequality.
Let ψn(u), n = 1, 2 ,... be the probability of ruin on or before the nth^ claim with initial capital u. Show that the Lundberg inequality implies that ψn(u) 6 e−Ru^ for all n = 1, 2 ,... and for all u > 0.
If μ = 1, show that ψ 1 (u) =
e−u 2 + ρ
and find ψ 2 (u).
Verify directly that ψn(u) 6 e−Ru^ for n = 1, 2 in this case.
