Solve the following questions. Kindly remember to explain them.

You have been asked to advise a sports magazine, as a consultant statistician. You have been asked to investigate the hypothesis that football managers in the Italian "Serie A", are dismissed more quickly than those in the English premier league. Each league has twenty teams, each with one manager. During the season the following events happened (at the end of the months indicated) to the twenty managers who started: Month Italy England 1 One dismissed One died One left of his own accord One left of his own accord One dismissed 6 Two dismissed One left of her own accord One died Two dismissed 11 Two dismissed One dismissed Hence there were thirteen of the original twenty managers still employed by the same club at the end of the season, for each of the two leagues. (i) Calculate the Kaplan-Meier estimate of the distribution function and its approximate variance for each league separately. [10] (ii) Comment on the hypothesis that Italian managers are dismissed more quickly than those in England. [2] [Total 12]The random variable X has a generalised Pareto distribution with density function, f(x;a) given by: f(xa) = a(a +1)(1,000)"x(1,000+ x)-@-2 x30 A sample of 100 values of X gives the following information: 100 log, (1,000 + x;) =750 i=l (i) Calculate the maximum likelihood estimate a of a. [6] (ii) Estimate the standard deviation of & . [3] [Total 9]