Weighting an Average to Minimize Variance

I wish there was a Strunk and White for mathematics.

While by no means logically incorrect, it feels inelegant to setup a problem using variables A and B in the first paragraph and solve for X and Y in the second (compounded with the implicit X==B, and Y==A).

This is just the observed variance. Which means that you assume that this will be the variance in the future.

Don’t make decisions for evolving systems based on statistics.

Insider info on the other hand works much better.

What a weird way to write the harmonic average.

----

Write v_i = Var[X_i]. John writes

    t_i = \frac{\prod_{j\ne i} v_j}{\sum_{k=1}^n \prod_{j\ne k} v_j}.

But if you multiply top and bottom by (1 / \prod_{m=1}^n v_m), you just get

   t_i = \frac{1/v_i}{\sum_{k=1}^n 1/v_k}.

No need to compute elementary symmetric polynomials.

If you plug those optimal (t_i) back into the variance, you get

    \min Var[\sum t_i X_i] = 1/(\sum_{k=1}^n 1/v_k) = H/n,

where `H = n / (\sum_{k=1}^n 1/v_k)` is the Harmonic Mean of the variances.