Given pdf is $\displaystyle f_y(y)=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}y^{a-1}(1-y)^{b-1}$ for $\displaystyle 0\le y\le1 $ $\displaystyle a>0$ $\displaystyle b>0$

a) Compute the moments about the origin $\displaystyle E(Y^m)$

b) Find the method of moments estimators for a and b (Using a sample of size n).

Attempt:

a)$\displaystyle E(Y^m) = \displaystyle\int_0^1 y^m \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}y^{a-1}(1-y)^{b-1}$

$\displaystyle =\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}*\beta(m+a, b)$

$\displaystyle =\frac{\Gamma(a+b)\Gamma(a+m)}{\Gamma(a+b+m)\Gamma (a)}$

b) Am i supposed to use part a to find the first and second central moments? I am not sure how to do this. Even so, by looking them up I try the following:

$\displaystyle E(Y) = \frac{a}{a+b} = m_1$

$\displaystyle E(Y^2) = \frac{a(a+1)}{(a+b)(a+b+1)} = m_2$

where m1 and m2 are the sample moments.

Basically when I try to solve this system of equations I get gross quadratics that I don't know how to simplify. Is there an elegant way of solving these, or if not, what is the best strategy for attacking these?