26  Pareto Distribution

A Pareto random variable \(X\) with parameter \(\alpha\) is a random variable with pdf \[f(x) = \frac{\alpha \theta^\alpha}{x^{\alpha + 1}},\qquad x > \theta\] Although there appears to be two parameters, only \(\alpha\) is a true parameter. The value of \(\theta\) must be set in advance.

The cdf of \(X\) is: \[F(x) = 1-\bigg(\frac{\theta}{x}\bigg)^\alpha, \qquad x > \theta\] and the sdf is: \[S(x) = \bigg(\frac{\theta}{x}\bigg)^\alpha, \qquad x > \theta\] Properties: * \(\text E[X] = \frac{\alpha\theta}{\alpha - 1}\) for \(\alpha > 1\) * \(\text{Var}(X) = \frac{\alpha\theta^2}{(\alpha-1)^2(\alpha-2)}\) * \(\text E[X^k] = \frac{\alpha\theta^k}{\alpha - k}\) for \(\alpha > k\) * The mgf of the Pareto distribution does not exist.