Past exam of the statistics course of the University of Oxford / 2021 / 1 / a / i / Solution

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 momentsIn particular, its mean is $\mu (\theta )=\theta /(\theta +1)$. Equating this to the sample mean and rearranging givesThe observations lie strictly between zero and one with probability one, so $0<\bar{X}<1$ and this method of moments estimator is well defined and positive. This is an instance of estimation of a beta shape parameter.