please help answer the first 3 questions.

1. John and Mary play a game as follows. John selects a real number 0 at random and writes it down on a piece of paper. Mary can ask John what 0 is. Each time when she does so, John tosses a biased coin secretly and tells Mary the number 0 - k if a head comes up (with probability p) and the number 0 + k if a tail comes up (with probability 1 - p) with k > 0. Mary knows the values of p and k, and has to guess the true value of O after she has asked John twice. Let X] and X2 be the two numbers John told Mary. Define X = (X1 + X2)/2. (a) Construct an unbiased estimator of 0, denoted by T, based on X . Find the variance of T. (b) Define W = X for X1 # X2; X1 + A for X1 = X2. i. Find the value of A such that W is an unbiased estimator of 0. Denote this unbiased estimator by U. ii. Find the value of A that minimizes the mean square error of W for esti- mating 0. iii. Is the estimator U in (i) relatively more efficient than T in (a)? 2. Suppose X1, X2, . . . , Xn constitute a random sample from a population following a N(u, 0) distribution, where the mean / is known and the variance 0 > 0 is unknown. (a) What is the distribution of (i - M)2, (b) Use the result from (a), show that Var(X?) = 202 + 4/20. You may use the fact that E(X; ) = M' + 3 0. (c) Show that 71 = = > X? - is a moment estimator of 0 by considering the n i= 1 second raw moment of X. (d) It is known that 72 = = > (Xi - M) is the maximum likelihood estimator of 1= 1 0. Which estimator is unbiased? 71, T2 or both? (e) Are 71 and 72 consistent estimators? (f) Find Eff(71, T2). Which estimator is more efficient? 3. Let X1 and X2 be i.i.d. exponential random variables with scale parameter 0, i.e., the pdf is f (x; 0) = -e-x/0, x>0. (a) Find a sufficient statistic for 0.(b) Show that both of the following estimators of 0 are unbiased: . X, the sample mean of X1 and X2; . 2X(1), where X() is the minimum of X1 and X2. (c) Calculate Eff(X, 2X(1)), the efficiency of X compared to 2X(1). Which estimator is more desirable? 4. Let {X1, X2, ..., X, } be a random sample from the Bin(n, p) distribution, where both n and p are unknown. (a) Derive the expressions for the moment estimators of n and p. (b) Using the estimator in part (a), estimate n and p based on the following observed sample: 6 5 7 11 5 10 5 9 12 9 7 7 9 10 8