Question

Statistics, finding the "Method of Moments" estimator for an unknown parameter theta.

Answer 1

In particular, my problem is this

Answer 2

The length of my understanding of this concept is that we are trying to find solutions to the equations\[\frac{ 1 }{ n }\sum_{i=1}^{n}X _{i}^{k}=E(X _{i}^{k}|\theta)\]for k=1,2,...,t
and that the left hand side are sample moments, while the right hand side are theoretical moments

Answer 3

I've come to the conclusion that i need to take another horrible integral...\[\frac{\Gamma(2\theta) }{ [\Gamma(\theta)]^2 }\int\limits_{0}^{1}X*X^{\theta-1}(1-X)^{\theta-1}dx=\frac{\Gamma(2\theta) }{ [\Gamma(\theta)]^2 }\int\limits_{0}^{1}X^{\theta}(1-X)^{\theta-1}dx\]but i can't seem to notice what kind of substitution i can use to do the integral, i feel like it can be manipulated to look like another distribution, although i don't know which one.
I am probably going to sleep right about now so any help i receive will not be responded to until morning, so i thank anyone in advance.