I am currently trying to compute / estimate the error of a function $f$ defined as
$$ f = \sum_{i=1}^N x_i \cdot \sum_{i=1}^N y_i $$
where I know the estimations for the $2N$ means $\bar{x}_i$ and $\bar{y}_i$ and their respective standard errors $e(\bar{x}_i)$ and $e(\bar{y}_i)$. I also know that $x_i$ is independent from all other $x_{j \neq i}$ and $y_{j \neq i}$ but that $x_i$ is dependent on $y_i$. My questions are now:
- Does a simple formula exist to calcualte $e(\bar{f})$ from the $e(\bar{x}_i)$ and $e(\bar{y}_i)$ that does not include the functional dependency of $x_i$ on $y_i$?
- In the limit of large $N$ can I simplify the situation by assuming that $x_i$ and $y_i$ are independent? Which would then give me something like $$ e(\bar{f}) = \sqrt{ \sum_{i=1}^N \left[ \left(\sum_{j=1}^{N} \bar{y}_j \right) e(\bar{x}_i)\right]^2 + \sum_{i=1}^N \left[ \left(\sum_{j=1}^{N} \bar{x}_j \right) e(\bar{y}_i)\right]^2 } $$ if my train of thoughts is right
Thank you in advance!