1. Suppose we compare three estimators, d1, (\"kg and 653, for a. We know that IE(11) = Edg = a Var(6z1) = 0% Var(c342) = 0%. If 6:3 = (16:1 + (1 (1)69, (a) Show that (343 is an unbiased estimator for 0:. (b) Show how to choose a so that variance of (3:3 is minimized. Assume (341 and (342 are independent. 2. A car manufacturer numbers their tanks uniformly over the interval (1,9). Suppose X1,X2, . . . ,Xn denotes a random sample of tanks. (a) Compute the bias and Show that X is a biased estimator of 9. (b) Find a function of X that is an unbiased estimator of 6. (C) Find the MSE when X is used as an estimator of 9. 3. Suppose Y1, Y2, Y3 denote a random sample from an exponential distribution with density function 379/9 ny)={ '9 ' E00 0, otherwise Consider the following estimators: 61_Y1 A Y+Y 62 12 2 9 _Y1+2Y2 3 2 64 = min(Y1,Yg,Y3) 95:17 (a) Which estimators are unbiased? (b) Among the unbiased estimators, which has the smallest variance? 4. Let Y1,}',...,Yn denote a random sample of size n from a population whose density is given by (21 \"a, OSySQ f(y) = 6 1 O, elsewhere where a > 0 is a known, xed value, but 9 is unknown. Consider the estimator g = max(Y1, Yg, . . . ,Yn). (a) Show that 6 is a biased estimator for 9. (b) Find a multiple of g that is an unbiased estimator of 9. A (c) Derive MSEUQ')