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

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 $Z$?

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 You should ask another question on the new issue instead of editing the existing one. – user28 Nov 11 '10 at 1:24 okay, reverted. will open a new question. thanks. – JWS Nov 11 '10 at 1:30

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