Suppose that $X_1,\ldots,X_n$ are normal with mean $\mu_1$; $Y_1,\ldots,Y_n$ are normal with mean $\mu_2$; and $W_1,\ldots,W_n$ are normal with mean $\mu_1+\mu_2$. Assuming that all $3n$ random variables are independent, with a common variance, find the maximum likelihood estimators of $\mu_1$ and $\mu_2$.
Solving for $\mu_1$ using $X_1,\ldots,X_n$ I got $\mu_1$ = the sample mean of $X$. Similarly, solving for $\mu_2$ using $Y_1,\ldots,Y_2$ I got $\mu_2$ = the sample mean of $Y$. However, I'm not sure if that's what this question meant, especially since I don't understand what the purpose of giving the mean of $W_1,\ldots,W_n$ is.