2. (a) Suppose X;~ N(0, 6), for i = 1, 2, 3, 4. Assume all these random variables are independent. Derive the value of k

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2 A Suppose X N 0 6 For I 1 2 3 4 Assume All These Random Variables Are Independent Derive The Value Of K 1 (82.31 KiB) Viewed 15 times

2. (a) Suppose X;~ N(0, 6), for i = 1, 2, 3, 4. Assume all these random variables are independent. Derive the value of k in each of the following. i. P(X₁ +8X2 < k) = 0.2119. (3 marks) ii. P · (2 x² < k) = = 0.99. (3 marks) iii. P (X₁ > (k(X² + X² + X²))¹/²) = 0.05. (4 marks) (b) Let {X₁, X2,..., Xn} be a random sample from the probability distribution with probability density function: Jo-¹ for 20≤x≤ −0 f(r;0)= to otherwise where > 0 is an unknown parameter. i. Derive the method of moments estimator of 0. (4 marks) ii. Is the estimator of 0 derived in part i. biased or unbiased? Justify your answer. Hint: You may use the fact that E(X) = E(X). (3 marks) iii. Given that the variance of the method of moments estimator of 0 in part i. is 02/(27n), check whether the estimator is a consistent estimator of 0. (3 marks)