Let X,..., Xn be an independent and identically distributed sequence of random variables from a population in (g(x | 0)| 0>0}, where g(x | 0) = 0x-1, 0 < x < 1. (a) [5 points] Show that the expectation of X is equal to 1 (b) [10 points] Show that the method of moment estimator for 00 is given by MM X 1-X Is this an unbiased estimator of 00? Hint: The function h(x) = is convex for all 0 < x < 1. (c) [10 points] Show that the maximum likelihood for 80 is given by ML n i=1 In(X) (d) [10 points] Show that {ge | 0> 0} is an exponential family and use the exponential family representation to calculate a sufficient statistic for 00. (e) [10 points] Using (d), argue that the estimator ODB==0ML is the UMVUE of 00- Hint: You may use that Eo (OML) = 10. (f) (10 points] Calculate the Cramr-Rao lower bound for any unbiased estimator of Does Windows ODB attain this bound? Does this contradict the claim in (e)? Go to Settings to activate Win Hint: You may use that Varo (OML) = (n-1)(n-20.