Suppose I draw $i$ random variables $X: x_1, x_2, x_3.... x_i$ independently from a normal distribution with an unknown mean $\mu$ and unknown variance $\sigma_1$. I do not observe the elements in $X$.
Rather, each draw becomes the mean of a new distribution. for each $x$, I observe $j$ independent draws from $N ~ (x_i, \sigma_{2}).$ This time, assume I know $\sigma_2$ but not each $x_i$.
I can compute my normally distributed expectation for each $x$. What I want to know is how to use that information to compute my expectation for $\mu$, $\sigma_1$.