Pareto Distribution (Percentile Points)
Calculate the percentile based on the lower and upper cumulative probability values of the Pareto distribution.
| \(\normalsize Pareto\ distribution\\ (1)\ probability\ density\\ \hspace{30px}f(x,x_{m},\alpha)={\large\frac{\alpha x_{m}^{\alpha}}{x^{\alpha + 1}}}\\ (2)\ lower\ cumulative\ distribution\\ \hspace{30px}P(x,x_{m},\alpha)={\large\int_{\hspace{15px}\small x_{m}}^{\small x}}f(x,x_{m},\alpha)dx={\large 1 - \left( \frac{x_{m}}{x} \right)^{\alpha} }\\ (3)\ upper\ cumulative\ distribution\\ \hspace{30px}Q(x,x_{m},\alpha)={\large\int_{\hspace{5px}\small x}^{\small \infty}}f(x,x_{m},\alpha)dx={\large \left( \frac{x_{m}}{x} \right)^{\alpha} }\\\) |
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