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In article <luke.722264309@barney>, luke@cs.city.ac.uk (Luke Whitaker) writes:
|> 
|> It seems that the only general rule for computing variances is for the sum or
|> difference of random variables (ie var(x+/-y) = var(x) + var(y)). What about 
|> products, reciprocals and other arbitrary combinations of random variables. 
|> I realise that this is non-trivial and that there is no simple answer as 
|> there is for sums and differences but there must be techniques to get a 
|> handle on this.
|> 
|> Could someone point me in the right direction, references etc for this problem.

An approximate answer to your question is given by the delta-method -- a method
for finding the asymptotic variance of f(x) for some smooth function f(.), as a 
function of var(x).
   In the simplest case, the delta method gives:
         approx var(f(x)) = f'(mu)^2 var(x)   
where f'(mu) is the first derivative of f evaluated at mu=E(x).
    For ratios of random variables and other functions, a more general 
(multivariate) version of the delta method may be used.  See either of the
references below for the details.
   
Ref'ns
  Bishop,YMM, Fienberg,SE, & Holland, PW (1975).  Discrete Multivariate Analysis:
Theory and Practice.  MIT Press: Cambridge, Mass. p486-502. 
  Serfling, RJ (1980).  Approximation Theorems of Mathematical Statistics.  Wiley:
New York.

 -- Roy St. Laurent
    Biostatistics
    Univ. of Mich.
    roy@sol.biostat.med.umich.edu
-- 
  -- Roy St. Laurent (roy@biostat.med.umich.edu)

"Even $10 calculators have sigma buttons.  [I] haven't seen one with an 
 M-estimator button, yet."         --Jerry Dallal in sci.math.stat

luke@cs.city.ac.uk (Luke Whitaker) writes:

>It seems that the only general rule for computing variances is for the sum or
>difference of random variables (ie var(x+/-y) = var(x) + var(y)). What about 
>products, reciprocals and other arbitrary combinations of random variables. 
>I realise that this is non-trivial and that there is no simple answer as 
>there is for sums and differences but there must be techniques to get a 
>handle on this.

>Could someone point me in the right direction, references etc for this problem.

A useful ref on the variance (and covariance) of products is

Bohrnstedt, G.W., \& Goldberger, A.S. (1969).  On the
exact covariance of products of random variables.  {\it Journal of the
American Statistical Association, 64}, 1439--1442.

They cite earlier work by Goodman which is also quite useful.

gary mcclelland
univ of colorado
mcclella@yertle.colorado.edu