Please answer this question

1. (30 points) Let X1, .... Xn be i.i.d. Exponential(1/0) random variables with common probability density function (pdf) f(x|0) = exp (-), x>0, 0>0. (a) (8 points) Derive the maximum likelihood estimator (MLE) of o based on X1, ..., Xn. Show that the MLE is unbiased for d. (b) (7 points) Show that the MLE has the smallest variance amongst all unbiased estimators of (c) (8 points) Consider testing the hypotheses Ho : 0 = 60 HA : 0 = 01, where do and 6 are numbers that are known to the experimenter, and 61 > do. Express the rejection region of the likelihood ratio test based on X1, ..., Xn in terms of the quantity ()_, X;) /60. You do not have to derive the test at any particular significance level o. Just give the form the rejection region must take. (d) (7 points) Suppose in part (c) we take n = 1, 60 = 1 and 61 = 2. Explicitly find the form of the rejection region of the test in part (c) which has significance level o for 0 0, P(Y > y) = e-".)