I apologise greatly in advance... I have been asked to show that the MLE for a normal distribution $X_i \sim N(\mu z_i, 1)$ is
$$ \hat{\mu} = \frac{\sum^{n}_{i=1} z_ix_i}{\sum^{n}_{i=1} z_i^2}$$
which is fine. I now want to prove that this estimator is unbiased. So clearly I have;
$$ E\left[ \hat{\mu} \right] = E\left[ \frac{\sum^{n}_{i=1} z_ix_i}{\sum^{n}_{i=1} z_i^2} \right] $$
And I know I'm looking to get this back to $\mu z_i$ but I'm just lost with this quotient and expectation. I don't see how I can carry the expectation through the quotient. A hint provided is to remember that the $z_i$ are constants. Which I guess is so at some point I will have $E\left[ x_iz_i \right] = z_i E\left[ x_i \right]$ but that's about it.
Again, so sorry that this is such a simple question!