Suppose $X_1,X_2,\ldots,X_n$ be a random sample of distribution with probability density function
$$f(x\mid\theta) = \theta x^{\theta-1},\quad 0\lt x \lt 1,\quad 0\lt \theta \lt \infty$$
how can i find MME of parameter $θ$?
My working:
$E[\theta x^{\theta-1}] = \bar X $
$\theta E[x^{\theta-1}]= \bar X$
$\theta = \bar X/E[x^{\theta-1}]$
Not sure whether this is correct.. Need some guidance on this...
Your working confuses random variables with numbers, and I suspect there is an error in the computation of the expectation. Note that, for every $k$, $$ E_\theta[X_k]=\int xf_\theta(x) \, \mathrm dx=\int_0^1\theta x^\theta \, \mathrm dx=\frac\theta{\theta+1}, $$ hence the first moment estimator $\hat\theta$ based on $(X_k)_{1\leqslant k\leqslant n}$ solves $$ \frac1n\sum\limits_{k=1}^nX_k=\frac{\hat\theta}{\hat\theta+1}, $$ that is, $$ \hat\theta=\frac{X_1+\cdots+X_n}{n-(X_1+\cdots+X_n)}. $$
