Question I:
We assume the classical case Poisson/Gamma.
We have:
St |Θ = θ ~ Poisson(θ)
Θ ~ Gamma(α , λ)
Calculate the marginal distribution St (all the intermediary steps are required) using the Moment Gener- ating Function (MGF).
Question II:
The distribution of Sn+1|Si1,…, Sin with probability density function f(x|x1,…, xn ) is called the predictive distribution of the random variable Sn+1 .
Show the following result (all the intermediary steps are required):
Bi,n+1 = E[µ (Θ)|Si1,…, Sin] = E[Si,n+1 |Si1,…, Sin].
Question III:
We assume the following mixing distributions: St |Θ ~ Poisson(Θ), with Θ ~ Gamma(α , T). The distribution functions are then:
(a) Find the posterior distribution of the heterogeneity parameter for the year T + 1, knowing S1 = s1,…, ST = sT , i.e. u(θ|S1,…, ST ) (all the intermediary steps are required).
(b) Find the predictive distribution of ST+1 knowing S1 = s1,…, ST = sT , i.e. Pr(ST+1|S1,…, ST ) (all the intermediary steps are required).
(c) Find the Bayesian (predictive) premium using the result in (a)
(d) Find the Bayesian (predictive) premium using the result in (b)
Question IV:
You are given:
(i) An individual insured has annual claim frequencies that follow a Poisson distribution with mean Λ .
(ii) An actuary’s prior distribution for the parameter Λ has probability density function:
(iii) In the first policy year, no claims were observed for the insured.
Determine the expected number of claims in the second policy year.
Question V:
You are given:
(i) The size of a claim has an exponential distribution with probability density function:
(ii) The prior distribution of Λ is an inverse gamma distribution with probability density function
for x ≥ 0.
For a single insured, two claims were observed that totaled 50. Determine the expected value of the next claim from the same insured.