Radiated seismic energies (in joules) of earthquakes above a catalogue completeness…

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

Radiated seismic energies

E

(in joules) of

n = 40

earthquakes above a catalogue completeness threshold

x_{\min} = 10^{12}\ \mathrm{J}

are modelled by the continuous power-law density

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

for

E \ge x_{\min}

. The sum of log-ratios is

\sum_{i=1}^{40} \ln(E_i/x_{\min}) = 54.0

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

\hat\alpha

and its standard error. ((b)) Estimate the fraction of these earthquakes with energy at least ten times

x_{\min}

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