Solution
= Solution
{c}
Differentiating the <cumulative distribution function> in part (ii) gives the <probability density function> $nt^{n-1}$ on $(0,1)$. Hence
$$
\mathbb E\left(\frac M\theta\right)=\int_0^1t\,nt^{n-1}\,dt=\frac n{n+1},\qquad
\mathbb E M=\frac n{n+1}\theta.
$$
Thus an <unbiased estimator> based on the sample maximum is
$$
\boxed{\widetilde\theta=\frac{n+1}{n}\widehat\theta.}
$$