Pareto Density Distribution

A Pareto Density Distribution is a power law probability distribution that ...

• See: Shape Parameter, Actuarial Science, Dirac Delta Function, Scale Parameter, Real Number, Power Law, Probability Distribution.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Pareto_distribution Retrieved:2015-3-19.
  • The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto, is a power law probability distribution that is used in description of social, scientific, geophysical, actuarial, and many other types of observable phenomena.
    • parameters =x_m > 0 scale (real)
      α > 0 shape (real)
    • support = [math]\displaystyle{ x \in [x_\mathrm{m}, +\infty) }[/math]
    • pdf = [math]\displaystyle{ \frac{\alpha\,x_\mathrm{m}^\alpha}{x^{\alpha+1}}\text{ for }x\ge x_m }[/math]
    • cdf = [math]\displaystyle{ 1-\left(\frac{x_\mathrm{m}}{x}\right)^\alpha \text{ for } x \ge x_m }[/math]
    • mean = [math]\displaystyle{ \begin{cases} \infty & \text{for }\alpha\le 1 \\ \frac{\alpha\,x_\mathrm{m}}{\alpha-1} & \text{for }\alpha\gt 1 \end{cases} }[/math]
    • median = [math]\displaystyle{ x_\mathrm{m} \sqrt[\alpha]{2} }[/math]
    • mode = [math]\displaystyle{ x_\mathrm{m} }[/math]
    • variance = [math]\displaystyle{ \begin{cases} \infty & \text{for }\alpha\in(1,2] \\ \frac{x_\mathrm{m}^2\alpha}{(\alpha-1)^2(\alpha-2)} & \text{for }\alpha\gt 2 \end{cases} }[/math]
    • skewness = [math]\displaystyle{ \frac{2(1+\alpha)}{\alpha-3}\,\sqrt{\frac{\alpha-2}{\alpha}}\text{ for }\alpha\gt 3 }[/math]
    • kurtosis = [math]\displaystyle{ \frac{6(\alpha^3+\alpha^2-6\alpha-2)}{\alpha(\alpha-3)(\alpha-4)}\text{ for }\alpha\gt 4 }[/math]
    • entropy = [math]\displaystyle{ \ln\left(\frac{x_\mathrm{m}}{\alpha}\right) + \frac{1}{\alpha} + 1 }[/math]
    • mgf =[math]\displaystyle{ \alpha(-x_\mathrm{m}t)^\alpha\Gamma(-\alpha,-x_\mathrm{m}t)\text{ for }t\lt 0 }[/math]
    • char = [math]\displaystyle{ \alpha(-ix_\mathrm{m}t)^\alpha\Gamma(-\alpha,-ix_\mathrm{m}t) }[/math]
    • fisher = [math]\displaystyle{ \begin{pmatrix}\frac{\alpha}{x_m^2} &-\frac{1}{x_m} \\ -\frac{1}{x_m} &\frac{1}{\alpha^2}\end{pmatrix} }[/math]

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Pareto_distribution#Definition Retrieved:2015-3-19.
  • If X is a random variable with a Pareto (Type I) distribution, then the probability that X is greater than some number x, i.e. the survival function (also called tail function), is given by :[math]\displaystyle{ \overline{F}(x) = \Pr(X\gt x) = \begin{cases} \left(\frac{x_\mathrm{m}}{x}\right)^\alpha & x\ge x_\mathrm{m}, \\ 1 & x \lt x_\mathrm{m}. \end{cases} }[/math] where x_m is the (necessarily positive) minimum possible value of X, and α is a positive parameter. The Pareto Type I distribution is characterized by a scale parameter x_m and a shape parameter α, which is known as the tail index. When this distribution is used to model the distribution of wealth, then the parameter α is called the Pareto index.