The attachments below contains three questions. Help in solving them.

Question 3.26 A random sample X1, .... X,, is taken from a normal distribution with mean / and variance oz. (i) State the distribution of Z ( x, - X)= 02 [1] It is decided to estimate the variance, o, using the following estimator: 6 2 = - 1 - E(x, - x)= n+b- where b is a constant. (ii) (a) Use part (i) to obtain the bias of 62. (b ) Hence, show that o is unbiased when b= -1. [3] (iii) (a) Show, using parts (i) and (ii)(a), that the mean square error of o- is given by: MSE(62) - 2(n-1) + (1+b)2 (n + b)2 (b) Determine whether the estimator, o, is consistent. (c) Show that the mean square error of o is minimised when b =1. [7] You may assume that the turning point is a minimum. (iv) Comment on the best choice for the value of b. [2] [Total 13]Question 3.22 Claims under a certain type of insurance policy occur as a Poisson process so that the number of claims arising from a policy in one calendar year is a Poisson random variable with mean # . In order to estimate / a random sample of n such policies is examined and it is found that no of these policies incurred no claims during the last full calendar year, n of them incurred one claim and the remainder incurred more than one claim. (i) Write down the likelihood of these observations and show that the maximum likelihood estimator / is a solution of the equation: m - (notn)ute"(nu + nu-m)=0 [ You do not need to verify that a maximum is attained.] [10] (ii) A random sample of 20 such policies yields no =8 and m =7. By using the tables of the cumulative Poisson distribution, or otherwise, to find a good starting approximation for / and then using trial and error (or a more sophisticated method), determine / correct to two decimal places. [7] [Total 17] Question 3.23 The claim amounts for a certain insurance company follow a Pareto distribution with probability distribution function given by: o250" (250 + x)-@-1 (i) Show that the maximum likelihood estimator for o is given by: a = n [In(250+ x;) -nIn 250 [4] 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 a . [5] [Total 9]