(Solved): 5. Let X,..., X be a random sample drawn independently from a distribution with probability ...

5. Let X?,..., X? be a random sample drawn independently from a distribution with probability density function x - ƒ (x; 0) = (k = (^² + = = ¹ ) 0² (1- ¹) 0² (1-0) ² for x = 0,1,2,..., k=1,2,... k - 1 This distribution has mean k(1 - 0)/0 and variance k(1 – 0)/0². (a) Find the maximum likelihood estimator Ô of 0. (You do not need to perform the second derivative test.) [5 marks] (b) Write down the score statistic U(X?,. Xn; 0) and show that the Fisher in- nk formation is given by I(0): = 0² (1-0) [5 marks] (c) Is the MVUE? Explain. [5 marks] (d) Let k = 2, n = 75 and x = 48. Obtain a 95% Wald confidence interval for 0. [5 marks] [20 marks]