Someone argues: "I have a record of samples of pure noise
PHYSICS · Multi-part · Fermi University Physics · Computational Physics · Electronics, Instrumentation & Communication · Scientific programming for physicists
Scientific Computing for Physicists, Ch Spectra with error bars: the fast transform, convolution and spectral estimation
Problem
Someone argues: "I have a record of N=4096 samples of pure noise. Its periodogram ordinates are exponentially distributed with mean equal to the true noise level S, so P(S^k>5S)=e−5=0.67%; with 2047 independent interior bins I expect about 14 bins above this threshold, which is what I see, so nothing is unusual. To be more confident in this null result, I now record N=65536 samples, sixteen times longer. This narrows each bin (better frequency resolution) and, since each periodogram value now averages over sixteen times more data, its variance must fall by a factor of 16. So a bin at 5× the mean level in the new record corresponds to an effective ratio of 5×16=80 at the old threshold probability, i.e. the same 0.67% chance now guards against a bin 80 times the mean rather than 5 times, which would mean a bin at only 5× the mean is far more significant with the longer record."
((a)) Identify the error in this argument.
((b)) State the correct probability that a bin of the N=65536 periodogram exceeds 5S.
((c)) In at most 80 words, explain why a longer record alone does not reduce this probability.