Address the questions below well

The claim amounts for a certain insurance company follow a Pareto distribution with probability distribution function given by: o 250" (250 + x)-@-1 (i) Show that the maximum likelihood estimator for o is given by: a = [4] [ In(250 + x; ) - nin 250 A random sample of 150 claims gives > In(250 + x; ) = 869 . (ii) Given that the estimator is asymptotically unbiased, obtain a 95% confidence interval for the parameter of. [5]The random variable X has a generalised Pareto distribution with density function, f(x;a) given by: f(x;a) = a(a+1)(1,000)"x(1,000 + x)-0-2 x>0 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 o of a. [6] (ii) Estimate the standard deviation of a . [3] [Total 9]Example 9.2.1. The following values were obtained during an investigation into the mortality of a. large group of lives: \"initial\" age exposed deaths to risk I E 3': 35 295,551 232 35 312,91? 295 3? 323,333 395 33 342,131 425 39 359,555 45'? 49 359,?91 59'1"Ir 41 359,521 552 42 335,513 T33 43 331,993 334 44 334,421 954 45 393,451 1,139 Test whether the mortality of this group of lives follows a. given table of which the following is an extract: age I 93, 35 9.99913 35 9.99 1991 3? 9.99 119? 33 9.99 1232 39 9.99 1339 49 9.99 15 51 41 9.99 1343 42 9.99 19 T3 43 9.99 2239 44 9.99 25 29 45 9.99 2343 45 9.9932 1 5 4'? 9.9935 23