A cumulative tail plot of city populations plots against

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 cumulative tail plot of city populations plots log⁡10P(X≥x)\log_{10}P(X \ge x) against log⁡10x\log_{10}x, where XX is population and the tail is modelled by a power law above some lower cutoff. Two points are read off the straight part of the plot: at x=105x = 10^5, P(X≥x)=0.20P(X \ge x) = 0.20; at x=106x = 10^6, P(X≥x)=0.025P(X \ge x) = 0.025. ((a)) Find the slope of the cumulative tail plot between these two points. ((b)) Convert that slope into the power-law exponent α\alpha, stating the relation you use and why the slope itself is not α\alpha.

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