Solve the questions attached below. Please ensure precision

For a certain class of policies issued by a large insurance company it is believed that the probability of each policy giving rise to any claims is 0.5, independently of all other policies. A random sample of 250 such policies is selected. (i) Determine approximately the probability that at least 139 of the policies in the sample will each give rise to any claims. (4] (ii) Suppose we do observe that 139 policies in our sample give rise to at least one claim. Use your answer to part (i) to determine whether this suggests at the 1% level of significance that the probability of any claims arising from a policy of this certain class is greater than initially believed. 131 [Total 7] A chi-square test of association for the frequency data in the following 2 x 3 table Factor A Al A2 43 BI 30 Factor B 40 50 B2 80 30 70 produces a chi-square statistic with value 4.861 and associated P-value 0.089. Consider a chi-square test of association for the data in the following 2 x 3 table, in which all frequencies are twice the corresponding frequencies in the first table: Factor A A2 43 BI 80 60 100 Factor B B2 160 60 140 (i) State, or calculate, the value of the chi-square test statistic for the second table. (2] (ii) Find the P-value associated with the test statistic in (i). [1] (iii) Comment on the results. (2] [Total 51Define the stationary distribution of a Markov chain. [2] A baseball stadium hosts a match each evening. As matches take place in the evening, floodlights are needed. The floodlights have a tendency to break down. If the floodlights break down, the game has to be abandoned and this costs the stadium $10,000. If the floodlights work throughout one match there is a 5% chance that they will fail and lead to the abandonment of the next match. The stadium has an arrangement with the Floodwatch repair company who are brought in the morning after a floodlight breakdown and charge $1,000 per day. There is a 60%% chance they are able to repair the floodlights such that the evening game can take place and be completed without needing to be abandoned. If they are still broken the repair company is used (and paid) again each day until the lights are fixed, with the same 60% chance of fixing the lights each day. (ii) Write down the transition matrix for the process which describes whether the floodlights are working or not. [1] (iii) Derive the long run proportion of games which have to be abandoned. 131 The stadium manager is unhappy with the number of games being abandoned, and contacts the Light Fantastic repair company who are estimated to have an 80% chance of repairing floodlights each day. However Light Fantastic will charge more than Floodwatch. (iv) Calculate the maximum amount the stadium should be prepared to pay Light Fantastic to improve profitability [4] [Total 10] The volatility of equity prices is classified as being High () or Low (L) according to whether it is above or below a particular level The volatility status is assumed to follow a Markov jump process with constant transition rates Opp = 4 and OHL =P. (i) Write down the generator matrix of the Markov jump process. [1] (ii) State the distribution of holding times in each state. [1] A history of equity price volatility is available over a representative time period. (iii) Explain how the parameters u and p can be cstimated. [2] Let , py be the probability that the process is in state jat time str given that it was in state i at time s (i, j - H, L), where 20. Let , py be the probability that the process remains in state i from time s to time sti