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(1) Define a Poisson process. [2] (ii) Prove the memoryless property of the exponential distribution. [2] Suppose there are three independent exponential distributions: X with parameter x Y with parameter y Z with parameter z (iii) (a) Demonstrate that min(X. Y.Z) is also an exponential distribution. (h) Give the parameter of this exponential distribution. [2] The arrivals of different types of vehicles at a toll bridge are assumed to follow Poisson processes whereby: Type of Vehicle Rate Motorcycle 2 per minute Car 5 per minute Goods vehicle 1.5 per minute The toll for a motorcycle is El. for a car $2 and for a goods vehicle $5. (iv) State the name of the stochastic process that describes the total value of tolls collected. [1] (V) Calculate the expected value of tolls collected per hour. [1] On the advice of a structural engineer, no more than two goods vehicles are allowed across the bridge in any given minute. If more than two goods vehicles arrive then some goods vehicles have to wait to go across. (vi) Calculate the probability that more than two goods vehicles arrive in any given minute. [2] (vii) Calculate the probability that exactly f4 in tolls is collected in a given minute. [4] [Total 14]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]