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Assignment IV

STATS 3ST3: ACTUARIAL MODELS IN NON-LIFE INSURANCE - Winter 2024

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)jSi1, ..., Sin] = E[Si;n+1 jSi1, ..., Sin].

Question III:

We assume the following mixing distributions: St|Θ ~ Poisson(Θ), with Θ ~ Gamma(α, τ ). 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).

(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:

πΛ (λ) = 0:5 [5e-5λ + 0:2e-0:2λ]

(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: