There is this problem:
Find the method of moments estimator of the parameter $\theta$ if you have a random sample of size n from the following distribution:
$f(x;\theta) = \theta ^2 xe^{-\theta x}$ for $0 < x$, zero otherwise; $0 < \theta$
So I attempted the problem and the integral looks absolutely atrocious and required multiple integration by parts. Is there an easier way to go about this problem?
Note that $X$ follows a Gamma distribution with $\alpha = 2$ and $\beta = \theta$ (see the parametrization at https://en.wikipedia.org/wiki/Gamma_distribution#Characterization_using_shape_%CE%B1_and_rate_%CE%B2; your $\beta$ might be $1/\theta$ instead depending on how your class teaches this). Hence, the mean is given by $$\dfrac{\alpha}{\beta}=\dfrac{2}{\theta}\text{.}$$ By the method of moments, this is set equal to the arithmetic mean $\bar{X} = \dfrac{1}{n}\sum_{i=1}^{n}X_i$. Hence, letting $\hat{\theta}$ be the method-of-moments estimator of $\theta$,
$$\dfrac{2}{\hat{\theta}} = \bar{X} \implies \hat{\theta}=\dfrac{2}{\bar{X}}\text{.}$$