Power Law & Pareto Distribution
Heavy tails, Zipf's law, and scale-free phenomena
Parameters
Exponent α
2.00
Min value x_min
1.0
Sample size
5000
Generate Samples
Real-world examples:
City sizes
α=1.16
Word freq
α=1.5
Earthquake
α=2.0
Wealth
α=2.5
Web links
α=3.0
Solar flares
α=1.8
Mean (finite if α>2):
—
Variance (finite if α>3):
—
Top 10% share:
—
%
Fitted α (Hill):
—
Pareto / power law
: P(X>x) = (x/x_min)^{-α+1}
PDF: p(x) = (α-1)/x_min · (x/x_min)^{-α}
Log-log plot
: straight line with slope -α.
Zipf's law
: rank-frequency follows p∝rank^{-1} ⟺ α=2. Scale-free: no characteristic scale — the distribution looks the same at every scale.