Suppose X1, ..., Xn are independent and identically distributed samples from a negative binomial distribution X ~ NBin(2, p). (a) Show that T = > X; is a sufficient statistic for p. (b) Show that the MLE for 0 = (1 -p) is 2n 2 T+2n (c) Show that the MLE for 0 is biased.(d) Show that the estimator S = 1 if X1 = 0 otherwise is an unbiased estimator for 0. (e) Use Rao-Blackwellization to show that 2n - 1 2n - 2 = T + 2n -1 T + 2n - 2 is an improved estimator for 0. Is it unbiased? (f) Show that the negative binomial distributions for a fixed number of failures r form an exponential family with respect to the sufficient statistic T. Use this to determine whether S is sufficient for p. (g) Compute the Cramer-Rao lower bound on the variance of the estimator you constructed in part (e). Is S an efficient estimator of 0? (h) Show that the estimators you constructed in parts (b) and (e) are consistent