Well, the question and my working out is shown below. I think that I know how to do the moment estimator maybe.. but I'm just having trouble with the integration part. I have a STRONG feeling that I'm doing it wrong. Can someone please help me?

Let X1 .. Xn be a random sample with pdf f(x) = e^-(x - theta), x> theta. Find the moment estimator and the maximum likelihood estimator of the parameter theta.