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A small island is holding a vote on independence. Two recent survey results are shown below: Poll Sample size Support for independence A 10 B 20 11 You should assume that the samples are independent. A politician is using a suitable uniform distribution as the prior distribution in order to estimate the proportion 0 in favour of independence. Calculate an estimate of 0 under the quadratic loss function. [3] A rival politician decides to use instead a beta distribution as the prior, with parameters a and B, where o = p. Determine the new estimate of Uunder the"all-or-nothing"loss function in terms of o [4] [Total 7] Claims X each year from a portfolio of insurance policies are normally distributed with mean 0 and variance to. Prior information is that O is normally distributed with known mcan u and known variance o-. Aggregate claims over the last n years have been a; for i = 1 to n, and you should assume that these are independent. Derive the posterior distribution of 0. [5] (ii) Write down the Bayesian estimate of 0 under quadratic loss. [1] (tii) Show that the estimate in your answer to part (ii) can be expressed in the form of a credibility estimate, including statement of the credibility factor Z. [2] [Total 8] (i) Explain what is meant by a saturated model. 121 (ii) State the definition of the scaled deviance in a fitting under generalised linear modelling. [1] (iii) (a) Define both Pearson and deviance residuals. (b) Explain how these two types of residuals are generally different. (9) State in which case they are the same. [5] [Total 8]