Average of Fractions

Thomas D. Schneider *

version = 1.01 of avfrac.tex 2014 Jan 19

*National Institutes of Health, National Cancer Institute, Center for Cancer Research, Gene Regulation and Chromosome Biology Laboratory, P. O. Box B, Frederick, MD 21702-1201. (301) 846-5581, email: schneidt@mail.nih.gov http://alum.mit.edu/www/toms/

Suppose we have n positive numbers, x1, x2, …xn.

We find the sum:

    ∑n
S =    xi
    i=1
(1)

If S > 0 we can formulate n fractions:

 f1  =  x1∕S                                     (2 )
 f2  =  x2∕S                                     (3 )

...                                              (4 )
 fn  =  xn∕S                                     (5 )
What is the average of the fractions?
       ∑
         ni=1fi
A  =   -------                                   (6 )
         n
   =   1-                                        (7 )
       n
No matter what the distribution of n fractions is, their average is always 1∕n!
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