Given a random sample $X_1,...,X_n$ that are $IID$ from an exponential population with a unknown parameter $\lambda>0$
The parameter of interest is $\theta=\frac{1}{2}\sqrt{\lambda}$
Im looking to find the method of moments estimator for $\theta$.
Without giving the answer out right could you please tell me if im on the right track $$\begin{aligned} E(X) &= \frac{1}{\lambda}\\ &=\bar{x}\end{aligned}$$ $$\begin{aligned} \frac{1}{2} \sqrt{\lambda} &= \theta \\ \sqrt{\lambda} &=2\theta \\ \lambda &= 4\theta^2\end{aligned}$$
And from here do i need to work out the mean of the sample of exponential random variables and which may perhaps use the properties of the gamma distribution? or do i just equate the mean of the sample $\bar{x}= E(X) = \frac{1}{\lambda}$ and then solve for the parameter of interest ($\theta$)?
Any pointers as to whether I'm on the right path at all much appreciated!