Given N scorers who each randomly independently choose a score from the uniform distribution on the real interval [0,1], show:
a. N∫0<x<1 x = N/2, then divide by N.
b. Reflection symmetry shows median has to be at 0.5.
c. N-1 ∫0<x<1 (x-0.5)2dx = 1/(12N).
d. The probability density the median lies at m is proportional to mH(1-m)H if N=2H+1. The mean location of the median is then
confirming answer (b). The mean squared noise in the median is
[All the integrals are Euler beta-function integrals.] This is
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