Provide step by step solution to the following questions.

An actuary is fitting the following linear regression model through the origin: Y, = Bx; te, e; ~ N(0.0 ) i=12. ...n (i) Show that the least squares estimator of / is given by: Ex,YA [3] (ii) Derive the bias and mean square error of & under this model. [4] [Total 7](i) Describe briefly the advantages and disadvantages of using a single figure index for comparing the mortality of two regions of a country. [4] (ii) The following data has been extracted from the 2001 census for the whole of a country with a developed economy and for two of its administrative regions. Region X Region Y Country Age Population Deaths in Population Deaths in Population Deaths in group at June 30 2001 at June 30 2001 at June 30 2001 (000s) (000s) (000s) 0-14 590 136 408 108 10,200 2,550 15-39 980 820 510 441 16,800 13,950 40-59 1,050 5,690 520 2,816 12,900 70,950 60-79 870 42,630 260 11,980 8,900 418,300 80+ 110 18,920 36 6,077 1,200 204,000 Total 3,600 68,196 1,734 21,422 50,000 709,750 (a) Calculate the crude death rate for each region and for the whole country. (b) Calculate the standardised mortality rate and the standardised mortality ratio for each region by reference to the country as a whole. (c) Comment on your results. [13] [Total 17]