Currently, I am studying for my exam in Statistics. We use Rice's book, so I am trying to do some of the exercises in this book. One of the exercises is as follows:
Suppose $X_1, \dots, X_n$ are i.i.d. with density function $ f(x \mid \theta) = e^{ - (x - \theta) } $, when $x \geq \theta $ , and $f(x \mid \theta ) = 0$ otherwise. Find the method of moments estimate of $\theta$.
I know that, in order to compute the method of moments estimate of $\theta$, one has to equate the theoretical average $ \bar{X} $ with the expectation. I don't know how to compute the expectation, though. I was thinking of finding it with a double integral:
$\mathbb E[f(x\mid\theta)] = \int_{0}^{\infty}\!\int_{\theta}^{\infty} e^{ - (x - \theta ) }\,dx\,d\theta,$ and then doing something similar in the region left of the y-axis. This, however, did not lead to the correct result.