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: […]