8. A random sample of size n. items is selected from the large number of items pro duced by a certain production line in one day. Find MLE of the ratio R, the proportion of defective items divided by the proportion of good items. 9. Suppose that n integers are drawn at random and with replacement from the integers 1,. . . , N. That is, each sampled integer has probabilityr 1 f N of taking on an}.r of the values 1, . . . ,N, and the sampled values are independent. (a) Find the methodofmoments estimator Ii." of N. (b) Find so?) and varpv). 1. Consider the following sequence of random variables Xn. Where each Xn takes one of two values 0 or n: Pr(X, = 0) = 1 and Pr(X, = n) = 1 -1. (a) Find lim, + E(X,), and (b) find limn-+ Pr(Xn 0 is some fixed number. 3. Suppose that X1, X2, X3, X4 denote a random sample of size 4 from a population with an exponential distribution with density function given by (1/0)e-x/0 if r > 0, f(x) = otherwise. (a) Let Y = vX1X2. Find a multiple of Y that is an unbiased estimator for 0. [You can use D(1/2) = VT. You might want to review I' function from STA 4321.] (b) Let W = VX1 X . .. x X4. Find a multiple of W that is an unbiased estimator for 02