Power Law Distribution

The power law distribution is also known as the Pareto distribution and scale-free distribution. A Random Variable is of power law distribution if its range is $\mathbb{N}$ (discrete) or $[c,\infty )$ (continuous) and

• Parameters
  • Discrete: $\alpha \in {\mathbb{R}}_{++}$
  • Continuous: $\alpha ,c\in {\mathbb{R}}_{++}$, $\beta =c^{\alpha }$
• CDF
  • Discrete: $P(X\ge k)=1/k^{\alpha }$
    • $F(k)=1-1/(k+1{)}^{\alpha }$
  • Continuous: $P(X>x)=\beta /x^{\alpha }$ for $x\ge c>0$
    • $F(x)=1-c^{\alpha }/x^{\alpha }$ for $x\ge c$
• PMF
  • $p(k)=1/k^{\alpha }-1/(k+1{)}^{\alpha }$
• PDF
  • $f(x)=\alpha c^{\alpha }/x^{\alpha +1}$
• Mean
  • Discrete: ${\sum }_{k=1}^{\infty }1/k^{\alpha }$
  • Continuous: $\frac{\alpha }{\alpha -1}$
• Variance
  • Discrete: ${\sum }_{k=1}^{\infty }2k^{1-\alpha }-k^{-\alpha }-{\left({\sum }_{k=1}^{\infty }{k}^{-\alpha }\right)}^2$
  • Continuous: $\frac{\alpha }{(\alpha -1{)}^2(\alpha -2)}$
• MGF
  • Discrete: $1+(e^t-1){\sum }_{k=0}^{\infty }{e}^{tk}(k+1{)}^{-\alpha }$

Note that when $\alpha$ is small, the “tail” $P(X\ge k)$ of the distribution decays slowly (slower than an exponential) as $k$ increases, and in some sense such a distribution has “heavy tails”.