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A sports league has two divisions {1,2) with Division 1 being the higher. Each season the bottom team in Division 1 is relegated to Division 2, and the top team in Division 2 is promoted to Division 1. Analysis of the movements of teams between divisions indicates that the probabilities of finishing top or bottom of a division differs if a team has just been promoted or relegated. compared with the probabilities in subsequent seasons. The probabilities are as follows: Finishing If promoted If relegated If neither promoted position previous season previous season nor relegated previous season Top 0. 1 0.25 0.15 Bottom 0.3 0.25 0.15 Other 0.6 0.5 0.7 (1) Write down the minimum number of states required to model this as a Markov chain. [ii) Draw a transition graph for the Markov chain. (Hii) Write down the transition matrix for the Markov chain. [2] (iv) Explain whether the Markov chain is: () irreducible. (b) aperiodic. [2] Team A has just been promoted to Division 1. (V) Calculate the minimum number of seasons before there is at least a 60% probability of Team A having been relegated to Division 2. (3] [Total 1 1]A certain proportion p of electrical gadgets produced by a factory is defective. Prior beliefs about p are represented by a Beta distribution with parameters o and B. A sample of n gadgets is inspected, and & are found to be defective. (i) Explain what is meant by a conjugate prior distribution. (ti) Derive the posterior distribution for beliefs about p. (3] (iii) Show that if X ~ Beta(a. B) with a > 1 then E- a+8-1 (3] 0- 1 (iv) It is required to make an estimate d of p. The loss function is given by Determine the Bayes estimate d* of p. [4] (v) Determine a parameter Z such that d' can be written as d* = Z x - 1(1-2)x_ where u is the prior expectation of I/p. 121 (vi) Under quadratic loss. the Bayes estimate would have been a+B+n Comment on the difference in the two Bayes' estimates in the specific case where o = =3, k - 2 and n - 10. [2] [Total 15]