What is the maximum likelihood estimator of the mean of two normally-distributed variables?

Question

Let $Z=(X+Y)/2$, where $X$ and $Y$ are independent normally-distributed random variables with known variances $\sigma^2_X$ and $\sigma^2_Y$ and unknown (and possibly different) means. Given a sample $x_1$ from $X$ and $y_1$ from $Y$, what is the maximum likelihood estimator of the mean of $Z$?

Answer 1

If $Z = \frac{X+Y}{2}$ then it must be that:

$Z \sim N(\frac{\mu_X + \mu_Y}{2} , \frac{\sigma_X^2 + \sigma_Y^2}{4})$

Thus, the mle of the mean of $Z$ given that we observe $z=(x_1 + y_1)/2$ is:

$\frac{x_1 + y_1}{2}$