Consider the family of probability density functions indexed by parameter $\theta \geq 1$ and given by:
$$f(x, \theta) = \frac{x}{\theta} e^{\theta - 1 - x}, x \geq \theta -1$$
For a random sample of size n, and justifying all steps:
Derive the method of moments estimator for $\theta$
$$E[X] = \int_{\theta -1}^\infty \frac{x^2}{\theta}e^{\theta - 1 - x}dx = ... = \frac{\theta^2 + 1}{\theta} \text{ (using integration by parts twice or WolframAlpha) and set this } = \overline{X}$$
$$\hat \theta_{MM} + \frac{1}{\hat \theta_{MM}} = \overline{X}$$
Is there a way to just isolate the estimator or is it okay to leave it like this?