Kindly solve appropriately
An industrial kiln is used to produce batches of tiles and is run with a standard firing cycle. After each firing cycle is finished, a maintenance inspection is undertaken on the heating element which rates it as being in Excellent, Good or Poor condition, or notes that the element has Failed. The probabilities of the heating element being in each condition at the end of a cycle. based on the condition at the start of the cycle arc as follows: START END Excellem Good Poor Failed Excellent 0.5 0.2 0.2 0.1 Good 0.5 0.3 0.2 Poor 0.5 0.5 Failed I (i) Write down the name of the stochastic process which describes the condition of a single heating element over time. [1] (ii) Explain whether the process describing the condition of a single heating element is: (a) irreducible. (bi periodic. [2] (iii) Derive the probability that the condition of a single heating element is assessed as being in Poor condition at the inspection after two cycles, if the heating element is currently in Excellent condition. 127 If the heating element fails during the firing cycle, the entire batch of tiles in the kiln is wasted at a cost of $1,000. Additionally a new heating element needs to be installed at a cost of 150 which will, of course, be in Excellent condition. (iv) Write down the transition matrix for the condition of the heating element in the kiln at the start of each cycle, allowing for replacement of failed heating elements. [2] (v) Calculate the long term probabilities for the condition of the heating element in the kiln at the start of a cycle. [4] The kiln is fired 100 times per year. (vi) Calculate the expected annual cost incurred due to failures of heating elements. [2] The company is concerned about the cost of ruined tiles and decides to change its policy to replace the heating element if it is rated as in Poor condition. (vii) Evaluate the impact of the change in replacement policy on the profitability of the company. [G] [Total 19]The following data is observed from n = 500 realisations from a time series: x, - 13153.32 , )(x, -7)2 - 3153.57 and (x -J)(x41 -X)-2176.03. i=1 i=1 Estimate, using the data above, the parameters , ay and o from the model X, - H=a (X ,-1 - H) +E, where a, is a white noise process with variance c. [7] (ii] After fitting the model with the parameters found in (i), it was calculated that the number of turning points of the residuals series e, is 280. Perform a statistical test to check whether there is evidence that &, is not generated from a white noise process. [3] [Total 10] The transition rules for moving between the three levels 0%, 35% and 50% of a No Claims Discount system are as follows: If no claim is made in a year, the policyholder moves to the next higher level of discount, or remains at the 50% level. When at the 0% or 35% level, the policyholder moves to (or remains at) the 0% level when one or more claims is made during the year. When al the 50% level of discount, the policyholder moves to the 35% level if exactly one claim is made during the year, or moves to the 0% level if two or more claims are made during the year. It is assumed that the number of claims X made each year has a geometric distribution with parameter q such that P(X = x) =q' (1-4). *=0,1,2. ... The full premium is 350. (i) (a) Write down the transition matrix. (b) Verify that the equilibrium distribution (in increasing order of discount) is of the form: ( kq- (2 q),kgil -9),4(1-9)3) for some constant k. Express & in tenns of q. [8]
