The claim amounts in a portfolio of insurance policies, X₁, X2,..., Xn, are assumed to follow a normal distribution with unknown mean 0 and known variance 1600. The prior information indicates that is normally distributed with mean 150 and variance 100. (a) Write down the likelihood for 0. (b) Show the posterior distribution of the parameter is the normal distribution N((nX+2400)/(n+16), 1600/(n+16)). (c) State the Bayesian estimate of under quadratic loss. (d) Show that the Bayesian estimate obtained in part (c) can be written in the form of a credibility estimate. (e) Suppose that the number of annual claims observed in a 3-year period are X₁ 200, X₂ = 300, X3 = 600, find the credibility factor and credibility estimate.
