First, assume you have variables $x$ and $y$. These variables are combined together to form a variable $t$ in the following way: $$t=ax + by$$ Where a and b are weights, and t is essentially a weighted sum
Now, assume you have two independent random variables: $$\sim N(μ_X,σ^2_X)$$ $$\sim N(μ_Y,σ^2_Y)$$
You want to combine $X$ and $Y$ in the same way that you combined $x$ and $y$. So you create a new random variable $$T = \alpha X + \beta Y$$
Now the question:
Assume that, to account for different levels of variance in $X$ and $Y$, you want to modify the equation for creating $T$ so that $$\alpha X + \beta Y$$ on average is as close as possible to $$\alpha μ_X + \beta μ_Y $$ Is there a way to weight this equation to make this discrepancy smaller?
Put another way, is there a value of $w$ such that $$T =w \alpha X + (1-w)\beta Y$$ is on average as close as possible to $$\alpha μ_X + \beta μ_Y $$