Average of Fractions

Thomas D. Schneider *

version = 1.01 of avfrac.tex 2014 Jan 19

https://alum.mit.edu/www/toms/

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

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!