Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics III / 7 / b / iii / Solution

Be careful about the index: the probability density function already contains a factor $x$, so $\mathbb{E}[X_i^r]=I_{r+1}$. The recurrence relation at $k=1$ yields $I_3=\theta I_1=\theta$. Therefore linearity of expectation givesThus $\hat{\theta }$ is an unbiased estimator.