Comment by KolenCh

This is equivalent to inverse variance weighting. For independent random variable, this is the optimal method to combine multiple measurements. He just used a different way to write the formula and connect that to other kinds of functions.

He also frames it as a different goal too: normally when we (as a physicist) talks about the random variables to combine, we think of it as different measurements of the same thing. But he didn’t even assume that: he’s saying if you want to have a weighted sum of random variables, not necessarily expected to be a measurement of the same thing (eg share same mean), this is still the optimal solution if all care is minimal variance. His example is stock, where if all you care is your “index” being less volatile, inverse variance weighting is also optimal.

As I’m not a finance person, this is new to me (the math is exactly the same, just different conceptually in what you think the X_i s are).

I wish he mention inverse variance weighting just to draw the connection though. Many comments here would be unnecessary if he did.