Method of Moment estimator for $\theta$ in a continuous random variable

Question

In the following question:

https://i.gyazo.com/910e9d364f724f4de26be71a11719af3.png

I have shown that $E[X] = \theta/(\theta-1)$. Now I have to deduce the method of moments estimator for $\theta$. I have equated my expectation formulae that I wrote above with sample mean equation $$1/N\sum_{i=1}^Nx_i$$ Is this correct because I don't know how to proceed from here?

Answer 1

Yes. You have $\mu=\theta/(\theta-1)$, this means the unknown $\theta=\mu/(\mu-1)$. Now replace $\mu$ with $1/N\sum_{i=1}^Nx_i$ i.e. the sample mean ($\hat{\mu}$) to get your estimator $\hat{\theta}=\hat{\mu}/(\hat{\mu}-1)$

Answer 2

Yes. The idea is correct. The estimator by moments $\hat{\theta}$ satisfies $$ \frac{\hat{\theta}}{\hat{\theta}-1}=\frac{1}{n} \sum_{n=1}^{N} x_n$$ So by solving for the estimator you get $$\hat{\theta}=\frac{\frac{1}{n} \sum_{n=1}^{N} x_n}{\frac{1}{n} \sum_{n=1}^{N} x_n-1}$$

Which is just $$\frac{\sum_{n=1}^{N} x_n}{\sum_{n=1}^{N} x_n-n}$$