1. (i) Maximize the following total profit function

= -3Q2 + 33Q - 72

(Determine that both the necessary and sufficient conditions for profit maximisationare satisfied)

ii) Evaluate the function in (i) at the desired critical value

A random sample X1. .... X,, is taken from a normal distribution with mean / and variance oz. (i) State the distribution of E(x, - x) [1] It is decided to estimate the variance, o, using the following estimator: 62 = . n +b E(x, - X)= where b is a constant. (ii) (a) Use part (i) to obtain the bias of &2. (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 - 04 (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]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)-@-2 *>0 A sample of 100 values of X gives the following information: 100 log, (1,000 + x;) = 750 i=1 (i) Calculate the maximum likelihood estimate a of a. [6] (ii) Estimate the standard deviation of a . (3] [Total 9]