Intuition behind power law distribution

I know that the pdf of a power law distribution is p(x)=α1xmin(xxmin)α

But what does it intuitively mean if, for example, stock prices follow a power law distribution? Does this mean that losses can be very high but infrequent?

Answer

This is an heavy tailed distribution, since the cdf is
F(x)=1(xxmin
So the probability to exceed x, (x/x_\min)^{1-\alpha} can be made arbitrarily close to 1 by the proper choice of \alpha. For instance, if one wants the probability to exceed 10^u x_\min to be at least 0.9, one should pick \alpha to be at most

1-\log_{10}(0.9)/u

a curve represented below, with the first axis being scaled by u, not by 10^u x_\min
R curve rendering of the above function

Attribution
Source : Link , Question Author : Thomas James , Answer Author : Xi’an

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