A survey records the peak X-ray flux of solar flares

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Probability, statistics and stochastic methods · Probability, statistics and stochastic methods

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

Problem

A survey records the peak X-ray flux of n=500n = 500 solar flares, in units in which the lower cutoff is xmin⁡=1x_{\min} = 1. Assume that flare fluxes above xmin⁡x_{\min} are drawn independently from a power-law density p(x)∝x−αp(x) \propto x^{-\alpha} for x≥xmin⁡x \ge x_{\min}, with no upper cutoff. For this sample, ∑i=1nln⁡(xi/xmin⁡)=300\sum_{i=1}^{n} \ln(x_i/x_{\min}) = 300. ((a)) Compute the maximum-likelihood exponent estimate α^\hat\alpha and its standard error σα^\sigma_{\hat\alpha}. ((b)) Using α^\hat\alpha, state with reasoning whether the mean flux and the variance of the flux are finite for this distribution. ((c)) A second survey under the same conditions records ten times as many flares (n=5000n = 5000). Using the largest-value scaling xmax⁡∼xmin⁡n1/(α−1)x_{\max} \sim x_{\min} n^{1/(\alpha - 1)} and the value of α^\hat\alpha from part (a), predict the factor by which the expected largest flare flux grows between the two surveys.

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