Estimate variance of a function given variances of variables

I'm given the mean and the standard deviation of N Gaussian random variables $$A$$, $$B$$, $$C$$, $$D$$... I compute the function $$f:=f(A, B, C, D...) = \frac { AB... }{ CD... }$$. How can I estimate the variance of the outcome of $$f$$?

EDIT I've edited my original post to reflect a more general case.

1 Answer

The general issue of how errors in a function of more than one random variable are passed through in a function of those variables is called "error propagation", which is a tag on this site. This page gives a pretty thorough explanation for the ratio case. Follow the error-propagation tag on this site, or do web searches for "error propagation" and "product" for the general case (as the $C$ or $D$ terms in the denominator can be written as products of the numerator with functions $1/C$, $1/D$, etc.)

Note that you get into trouble if any of the random variables in the denominator have mean values close to 0, and that correlations among the random variables can make it difficult to express a simple formula.

If errors in $X$ and $Y$ are approximately proportional to their values, then you can analyze the logarithm of the ratio, and the complicated product calculations become a simple additive case, $log(f)= log(X) - log(Y)$ for the original question about the ratio $X/Y$. Then the standard law for the variance of a sum/difference of random variables applies. For the more general case in the edited question, logarithms similarly simplify if appropriate.

• I've edited my post to represent a more general case, that is there can be products of variables along with ratios. May 17 '15 at 20:01
• I've found some information about error propagation that covers my question. Thank you a lot for telling me the correct term to search. May 17 '15 at 20:32