Kindly answer all in details with explanation

1 (i) Explain with the aid of formulae what is meant by the term 'Standardised Mortality Ratio' (ii) For a certain year, the estimated mid-year population of women in England and Wales and the number of deaths from cancer were as follows: Exact Estimated Number of ages population deaths 0-35 12,039,500 939 35-50 4,741,100 3,925 50-65 4,192,400 15,549 65-80 3,471,900 29,337 over 80 1,148,600 17,544 For the same year, the corresponding data for women in a certain district were as follows: Exact Estimated Number of ages population deaths 0-35 83,902 5 35-50 29,970 31 50-65 30,102 98 65-80 19,220 149 over 80 5,799 82 Calculate the Standardised Mortality Ratio with respect to cancer for women in this district for the year in question, using the national rates as the standard. 2 The following data refers to permanent assurances, males, in 1988-89 at durations 2 years and over. age-group Smokers Non-smokers (nearest Actual 100A/E Actual 100A/E ages) deaths by AM80 deaths by AM80 ultimate ultimate 76 75 98 30 61 Total 594 91 109 53 Source C.M.I. Bureau (i) Suppose that, at all ages, the force of mortality of non-smokers is ow times the force of mortality of AM80 ultimate, whilst that of smokers is as times the force of mortality of AM80 ultimate. Find approximate 95% confidence intervals for on and as. (ii) Suppose that, at all ages, the force of mortality of the combined group (of smokers and non-smokers together) is ac times the force of mortality of AM80 ultimate. Find an approximate 95% confidence interval for ac. 3 Let # ~ Poisson(mE), where E is a constant and where m takes the (unknown) value m. Show that m = 0/E is the maximum likelihood estimator of m, and also that it is an unbiased and efficient estimator of m.The random variable Y is defined by: N Y = EX; i=1 where the X;'s are independent Bin(n, p) random variables and N is a Poisson random variable, independent of the X; 's, with mean 1. (i) Prove from first principles that the moment generating function of Y is given by: My (1) = exp A [(q + pe')" -1]} [4] (ii) Hence derive expressions for the mean and variance of Y. [4] [Total 8]