I want to prove that an estimator in biased. For a big n, like $10^{13}$ we define a distribution of values $x_i \in (0,1]$ for $i \in [n]$. We want to estimate with some error $\epsilon$ the value of:
$$\mu = \sum_{i}^n \log x_i$$
The distribution of $x_i$ the values of $x_i$ is not uniform on the interval: there are some (like $10^2$) values that are very close to one, and the rest are almost zero.
We are given $m$ samples of $x_i$ with uniform distribution, that is: $p(x_i)=p(x_j)$ for $j\neq i$.
Then, I compute $\overline{\mu}=n \times \frac{1}{m}\sum_{i=1}^m \log x_i$.
I want to show that $\overline{\mu}$ does is not the right estimator for $\mu$, and I think it is biased, i.e.: $$E[\overline{\mu}] \neq \mu$$.