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Pareto Distribution Percent Point (Quantile) Calculator

Find the Pareto distribution percent point x = xm / (1 − p)^(1/α) in closed form from probability p, scale xm and shape α, with PDF, CDF, mean and variance plotted.

Input

Find the percent point (quantile) of the Pareto distribution (Type I). Enter a probability with the scale xm and shape α.

A value between 0 and 1 (e.g. 0.95)

Probability type

Whether p is a lower tail or upper tail probability

Lower bound of the distribution. A positive value (e.g. 1)

Controls the tail heaviness. A positive value (e.g. 3)

Result

Percent point x for lower probability 0.95

2.71441762

Scale xm

1

Shape α

3

Mean

1.5

Variance

0.75

Lower probability F(x)

0.95

Upper probability 1 − F(x)

0.05

Density f(x)

0.05526047

Probability density f(x)

Cumulative distribution F(x)

How it works

  • The Pareto distribution (Type I) is a continuous distribution with support x ≥ xm, where the scale xm sets the lower bound and the shape α controls how heavy the tail is.
  • For a lower tail probability p the percent point has the closed form x = xm ÷ (1 − p)^(1 ÷ α).
  • When an upper tail probability is given it is converted internally to the lower probability 1 − p before applying the same formula.
  • The cumulative distribution is F(x) = 1 − (xm ÷ x)^α and the density is f(x) = α xm^α ÷ x^(α + 1) for x ≥ xm.
  • The mean is finite only when α is greater than 1, equal to α xm ÷ (α − 1); otherwise it diverges.
  • The variance is finite only when α is greater than 2, equal to xm² α ÷ ((α − 1)² (α − 2)); otherwise it diverges.
  • Enter a probability p strictly between 0 and 1, and positive values for both the scale xm and the shape α.

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