A geological survey records the volumes (in m ) of landslides in a mountain range

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

A geological survey records the volumes VV (in m3^3) of n=50n = 50 landslides in a mountain range, all above a completion threshold xmin⁡=1000 m3x_{\min} = 1000\ \mathrm{m}^3 below which small slides are not reliably logged. The volumes are modelled as drawn from a continuous power-law density p(V)∝V−αp(V) \propto V^{-\alpha} for V≥xmin⁡V \ge x_{\min}. The sum of the log-ratios is ∑i=150ln⁡(Vi/xmin⁡)=62.3\sum_{i=1}^{50} \ln(V_i/x_{\min}) = 62.3. ((a)) Find the maximum-likelihood estimate α^\hat\alpha and its standard error. ((b)) Using α^\hat\alpha, estimate the fraction of landslides with volume at least 10,000 m310{,}000\ \mathrm{m}^3 (ten times xmin⁡x_{\min}).

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