Q 6. In the following, let the random vector X represent all the past data and let Xn+1 represent the next observation. Let g(X) be any function of the past data. (a) Prove that the following is true: B{{Xn+1 g(x)}} =E {{Xn+1 = E(Xn+1|X)/?} +E{1E (Xn+1 / X) g(x))"} where the expectation is taken over (X+1, X). (b) Show that setting g(x) equal to the Bayesian premium (the mean of the predictive distribution) minimizes the expected squared error, E{{Xn+1 g(x))"} (e) Show that, if g(x) is restricted to be a linear function of the past data, then the expected squared error is minimized by the credibility premium. a Q 6. In the following, let the random vector X represent all the past data and let Xn+1 represent the next observation. Let g(X) be any function of the past data. (a) Prove that the following is true: B{{Xn+1 g(x)}} =E {{Xn+1 = E(Xn+1|X)/?} +E{1E (Xn+1 / X) g(x))"} where the expectation is taken over (X+1, X). (b) Show that setting g(x) equal to the Bayesian premium (the mean of the predictive distribution) minimizes the expected squared error, E{{Xn+1 g(x))"} (e) Show that, if g(x) is restricted to be a linear function of the past data, then the expected squared error is minimized by the credibility premium. a