A random sample of size $n_1$ is to be drawn from a normal population with mean $\mu_1$ and variance $\sigma^2_1$.
A second random sample of size $n_2$ is to be drawn from a normal population with mean $\mu_2$ and variance $\sigma^2_2$. The two samples are independent.
What is the maximum likelihood estimator of $α = \mu_1 − \mu_2$?
Assuming that the total sample size $n = n_1 + n_2$ is fixed, how should the $n$ observations be divided between the two populations in order to minimise the variance of $\hatα$?
I know how to find the MLEs of $\mu_1$ and $\mu_2$, but I don't know how to use these to find $α$ for the first part of this question. I don't even know where to start on minimising the variance.