Suppose we have
$$A_i \sim N( \mu_a, \sigma_a)$$ $$B_i \sim N( \mu_b, \sigma_b)$$
Where $A_i$ and $B_i$ are i.i.d. respectively, where $i = 1 \ldots n$, We are interested in $$\frac{ \sum_{i=1}^{n}{ \frac{A_i}{ B_i} } }{n} $$ and $$ \frac{\sum_{i=1}^{n}{A_i} }{ \sum_{i=1}^{n}{B_i} } $$ Obviously these 2 values wont be the same except for very unique cases. However, from looking at these it seems to me that there must be some sort of statistical link between the 2 overall values.
I am by no means suggesting that there is a direct proportionality between the 2, however it appears that there may be some tenuous link up to a certain confidence level where you can say it is true that the 2 will differ by 'some factor' for a given variance or maybe it is something else that determines this 'factor'.
what I would be interested in knowing is:
- If there is a link between the 2
- If so, then is it perhaps determined by the variance of the denominator, or the nominator, or both
- Does it depend on the type of distribution, or its support
- If there was a factor of sorts linking this, and what the confidence level might be.
I would be interested in seeing if there was a general rule similar to something like the Rule Of Three. I'd be extremely grateful if anyone could find something of this nature!
Ps. I couldn't find theory on this on the web, however if there is something like this out there then could you please link it below and I do realise this is an awful lot to consider, I'd be surprised to get an actual answer or validation on this in honesty.