= Solution
{c}
For the <beta distribution> with parameters $(\theta,1)$, the <Euler beta function> gives $B(\theta,1)=1/\theta$. Its <probability density function> is therefore $\theta x^{\theta-1}$ on $(0,1)$. Direct <integration> gives the <raw moments>
$$
\mathbb E_\theta X^r=\theta\int_0^1x^{\theta+r-1}\,dx=\frac{\theta}{\theta+r},\qquad r> -\theta.
$$
In particular, its <mean> is $\mu(\theta)=\theta/(\theta+1)$. Equating this to the <sample mean> and rearranging gives
$$
\theta=\bar X(1+\theta),\qquad
\boxed{T_1=\frac{\bar X}{1-\bar X}}.
$$
The observations lie strictly between zero and one with <probability> one, so $0<\bar X<1$ and this <method of moments (statistics)> <estimator> is well defined and positive. This is an instance of <estimation of a beta shape parameter>.
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