File sizes (in MB) on a data-centre network are logged for files above a floor of…

PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Mathematical Methods for Physics · Data analysis and model fitting in the laboratory

Many Parts, One Behaviour, Ch Heavy tails: power laws and how to measure them

Problem

File sizes

S

(in MB) on a data-centre network are logged for

n = 80

files above a floor of

x_{\min} = 1\ \mathrm{MB}

(system files below this are excluded from the log), and modelled by the continuous power-law density

p(S) \propto S^{-\alpha}

for

S \ge x_{\min}

. The sum of log-ratios is

\sum_{i=1}^{80} \ln(S_i/x_{\min}) = 97.5

. ((a)) Find the maximum-likelihood estimate

\hat\alpha

and its standard error. ((b)) Estimate the fraction of files with size at least

50\ \mathrm{MB}

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