Sir please solve this

4. [20 marks] Given A, k > 0, a random variable X is said to follow the Weibull distribution with parameters (), k) if its probability density function is given by *)-tax), 120. i) For k 2 1, show that the moment generating function of X is given by Eletx ] = Emr ( 1 + *), ter where I(z) = fo x* le "dx, z > 0 and I(z + 1) = =[(2). Hint: Use the fact that et = Zin=0 n!. ii) Show that the n-th moment is given by AT ( 1 + 22 ) Hint: Assume that one can differentiate under the infinite sum. iii) In the following, we suppose the parameter k 2 1 is fixed. a) Compute the method of moment estimator and the maximum likeli- hood estimator of A, which are denoted by Amme and Amie. b) Show that (Amle)" is an unbiased estimator of Ak. c) Show that Bias(Amle) 5 0. d) Compute the Fisher information for 1. e) Compute an approximate two sided 95% confidence interval for the parameter 1k. You may use the following R-output > qnorm (0 . 975, 0, 1) [1] 1.959964 > qnorm (0 . 95 , 0, 1) [1] 1.644854