Let X1, ..., Xn be a random sample from the exponential distribution with density function being f(x) = ex , where is positive and unknown and n is large enough (n 30). (a) Please derive the 100(1)% confidence interval for , using the pivotal quantity method. (Please include the derivation of the pivotal quantity and the derivation of the confidence interval for full credit.) (b) Please derive the test for H0 : = 0 versus Ha : > 0 at the significance level using the pivotal quantity method (Please include the derivation of rejection region for full credit.) (c) Please derive the likelihood ratio test for H0 : = 0 versus Ha : 6= 0