# Intuition behind power law distribution

I know that the pdf of a power law distribution is $$p(x) = \frac{\alpha-1}{x_{\text{min}}} \left(\frac{x}{x_{\text{min}}} \right)^{-\alpha}$$

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?

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This is an heavy tailed distribution, since the cdf is $$F(x) = 1 - \left( \dfrac{x}{x_\min} \right)^{1-\alpha}$$ 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 $10^u x_\min$...