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Questions from an actuarial statistics m.phil. Exam. Topics include defining cumulant generating functions and cumulants, finding mean and variance of poisson distributed random variables, calculating the probability generating function of the total number of claims, and modeling a no claims discount system using a discrete-time markov chain.
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Thursday 31 May 2001 9 to 11
Attempt any THREE questions. The questions carry equal weight.
1 Define the cumulant generating function and the cumulants of a random variable Y. Show that the first and second cumulants are the mean and variance respectively. Let T =
i=1 Vi^ where^ N^ has a Poisson distribution with mean^ ν, and^ V^1 , V^2 ,...^ are independent and identically distributed random variables, independent of N. Show that the cumulant generating function of T is κT (t) = ν(MV (t)−1) where MV (t) is the moment generating function of Vi. Hence or otherwise find ET and varT in terms of ν and moments of Vi.
In a particular region, the number M of floods in a year has a Poisson distribution with mean ν. The total amount of claims arising from the ith^ flood is Si =
∑Ni j=1 Xij^ , i^ = 1 ,... , M, where N 1 , N 2 ,... are independent Poisson random variables with mean λ, Xij denotes the jth^ claim from flood i, the Xij ’s are independent with EXij = μ, varXij = σ^2 for all i, j. Assume M, {Ni}, {Xij } are independent, and let S be the total amount of claims arising from all the floods occuring in one year in this region. Find ES and varS.
Find the probability generating function of the total number N of claims in one year and determine whether or not N has a Poisson distribution.
Suppose that, independently for each claim, there is probability p that it is found not to be valid. Find the distribution of the number of valid claims arising from the ith flood. Find the expectation of the total amount paid in a year on valid claims.
2 In a classical risk model, claims arrive in a Poisson process rate λ > 0, the claim sizes have density f (x) and the premium income rate is c > 0. Let φ(u) be the probability of never being ruined when the initial capital is u > 0. Show that
φ′(u) =
λ c
φ(u) −
λ c
∫ (^) u
0
φ(u − x)f (x)dx.
Suppose that f (x) = 3e−^4 x^ + 12 e−^2 x, x > 0, and that the relative safety loading is ρ = 3/5. Show that
φ′′′(u) + 4φ′′(u) + 3φ′(u) = 0.
Given φ(0) = (^) 1+ρρ , find φ(u).
