Power

• $\mathrm{min}$: minimum value (lower bound, $\mathrm{min} > 0$)
• $\mathrm{shape}$: shape parameter ($\alpha > 0$, controls tail heaviness; smaller values = heavier tails)

• Formation: follows a power-law relationship where large values are rare but possible.
• Origin: historically called Pareto distribution after Vilfredo Paretos work on wealth distribution.
• Rename Motivation: renamed to $\underline{\operatorname{Power}}$ to reflect its connection with power-law.
• Properties: exhibits scale invariance and extremely heavy tails.
• Applications: wealth distribution, city population sizes, word frequencies, earthquake magnitudes, website traffic.
• Characteristics: infinite variance for many parameter values; extreme outliers are common.
• Caution: traditional variance-based estimators completely fail; robust estimators are essential for reliable analysis.