Provide clear and precise answers.
Consider two random variables Y and Y with E[X] = 2, V[X] = 4, A[Y] =-3, VIY] =1, and Cov[X, Y] = 1.6. Calculate: (a) the expected value of SX + 201. (b) the correlation coefficient between X and Y. the expected value of the product XY. (d) the variance of X - Y. [4] In a medical study conducted to test the suggestion that daily exercise has the effect of lowering blood pressure, a sample of eight patients with high blood pressure was selected. Their blood pressure was measured initially and then again a month later after they had participated in an exercise programme. The results are shown in the table below: Patient 2 3 4 5 6 7 8 Before 155 152 146 153 146 160 139 148 After 145 147 123 137 141 142 140 138 (1) Explain why a standard two-sample /-test would not be appropriate in this investigation to test the suggestion that daily exercise has the effect of lowering blood pressure. [1] (ii) Ferform a suitable r-test for this medical study. You should clearly state the null and alternative hypotheses. [7] [Total 8]Consider the following discrete distribution with an unknown parameter p for the distribution of the number of policies with 0, 1, 2, or more than 2 claims per year in a portfolio of n independent policies. number of claims 2 more than 2 probability 2p D 0.25 p 1-3.25 p We denote by Xo the number of policies with no claims, by X, the number of policies with one claim and by X, the number of policies with two claims per year. The random variable X = Xn+ X, + X, is then the number of policies with al most two claims. (i) Derive an expression for the maximum likelihood estimator / of parameter p in terms of X and n. [5] (ii) Show that the estimator obtained in part (i) is unbiased. [3] The following frequencies are observed in a portfolio of a - 200 policies during the year 2012: number of claims 2 more than 2 observed frequency 123 58 13 A statistician proposes that the parameter p can be estimated by p=58/200=0.29 since p is the probability that a randomly chosen policy leads to one claim per year. (iii) Estimate the parameter p using the estimator derived in part (i). (iv) Explain why your answer to part (iii) is different from the proposed estimated value of 0.29. [2] An alternative model is proposed where the probability function has the form number of claims 2 more than 2 probability 2F 0.25 p 1-3.25p (v) Explain how the maximum likelihood estimator suggested in part (i) needs to he adapted to estimate the parameter p in this new model. (1] (vi) Suggest a suitable test to use to make a decision about which of the two models should be used based on empirical data. [1] [Total 13]
