A colleague put the mess in my head following a discussion about what he calls the "best weighting" from a statistic point of view when we treat 2 random variables.
He told me that if we have the sum of random variables : $Z=X+Y$, we can show that the best estimator is :
$$\sigma_z^2 = \alpha \sigma_x^2 + (1-\alpha)\sigma_y^2$$ wih $$\alpha=\dfrac{\sigma_x^2}{\sigma_x^2+\sigma_y^2}$$
Where do you come from this reasoning ? and how to prove it ?
Is this reasoning only true when we consider $X$ and $Y$ as Gaussian variables ?
