2

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.

Jack Pedley
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  • Sorry you're totally right, I realise that it didn't make sense. Hopefully the edited version is clearer. – Jack Pedley Aug 24 '15 at 16:47
  • Much clearer. I've removed my comments. – joriki Aug 24 '15 at 17:07
  • I think this could be a quite difficult problem to study... For example, if you take mean zero and variance 1, for both $A$ and $B$, then the ratio $A_i/B_i$ is a standard Cauchy (so the average is also a standard Cauchy) and the ratio of sums is also Cauchy (I think). So even studying their correlation does not make sense... – M Turgeon Aug 24 '15 at 17:09
  • Some useful insight could be gained in the case where the standard deviations are small with respect to the means by expanding with respect to $\Delta B_i=B_i-\bar B$ and calculating the means, variances and covariance of the two expressions up to second order. – joriki Aug 24 '15 at 17:15

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