Two measurements $X$ and $Y$ are independently drawn from the same distribution with mean $\mu$ and variance $\sigma^2$, and a weighted sum $S = wX + (1 − w)Y$ is computed.
- What is the expectation and variance of $S$?
- What is the value of $w$ that minimizes the variance of $S$ and the min value of variance of $S$?
I think this is much simpler than I'm making it out to be.
$$\operatorname{E}(S)=w\operatorname{E}(X)+(1-w)\operatorname{E}(Y)=\mu$$ $$\operatorname{Var}(S)=w^2\operatorname{Var}(X)+(1-w)^2\operatorname{Var}(Y)=\sigma^2(w^2+(1-w)^2)=\sigma^2(2w^2-2w+1)$$ These two follow from the linearity of expectation and because the variance is additive when $X$ and $Y$ are independent. You can answer the second one by taking derivatives.