I need to find the Method Moment Estimator of parameter $\theta$ based on a random sample $X_1…X_n$ with the following pdf:
$f(x;\theta)=\theta^2x^2e^{-{\theta}x}$; $0<x$, zero otherwise; $0<\theta$
I am completely stuck on this. I know that the answer is supposed to be $\frac{2}{\bar{X}}$. I don't have the slightest idea how to arrive at that answer.
I am still new to this MME thing, but so far as I can tell, this pdf doesn't match any 'set' distribution, so I can't just plug in known moments. I have to calculate using integration. But how do I integrate this pdf?
If I plug it into WolframAlpha's integral calculator, it gives me the solution $\frac{e^{-x\theta}(-{\theta}x({\theta}x+2)-2)}{\theta}$. When I try to integrate by parts (I don't feel like typing out the entire process, because I did this in pen and paper), I get something that looks like $-\theta^2e^{-{\theta}x}(\theta^2x^2+2{\theta}x+2)$. Either way, this doesn't lead me anywhere close to the answer.
So any recommendations for how to solve this problem?