Consider I have $2$ returns time series random variables $r_{i,t}=\ln{\left(\frac{p_{i,t+1}}{p_{i,t}} \right)}\stackrel{(i.d.)}{\sim}\mathcal{N}(\mu_i,\sigma_i)$.
Let's now compute a weighted average:
$$ R_t=\frac{\sum_{i=1}^{2}{w_{i,t}r_{i,t}}}{\sum_{i=1}^{2}{w_{i,t}}} $$
- How would the distribution of $R_t$ look like if $w_{i,t}=p_{1,t}$ i.e. if we are doing a price-weighted average? Can we say something about $\mathrm{E}[r]$ and $\mathrm{Var}[r]$ for example?
- And if instead $w_{i,t}=w=0.5$ i.e. we are compute a simple average?
Any insight is really appreciated.