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The standard Pareto distribution is a Pareto distribution with shape parameter 1 and scale parameter 1.

Probability Density Function

f ( x ) = 1 x 2
F ( x ) = 1 − 1 x
S ( x ) = 1 x
h ( x ) = 1 x

Support

1 ≤ x < ∞

Mean

Variance

Example
Suppose U has a standard uniform distribution. Then 1/U has a standard Pareto distribution.
Suppose a probability p has a standard uniform distribution. Then 1 + odds(p) has a standard Pareto distribution.
Suppose Y has a standard logistic distribution. Then 1 + eY has a standard Pareto distribution.

X ∼ Standard Pareto

Chart of the standard Pareto distribution Chart area for displaying the standard Pareto pdf, cdf, survival, hazard, visualization, and simulation

E(X) = , Var(X) =

Note that the standard Pareto distribution is heavy-tailed, and that the mean and variance are undefined.

Note that the standard Pareto distribution is heavy-tailed, and that the mean and variance are undefined.

The graph above displays the survival function S(x) = P(X > x) = 1 - F(X), where F(x) is the cumulative distribution function (cdf).

Survival functions are used in survival analysis, a branch of statistics concerned with the expected duration until an event occurs such as death or the failure of a mechanical system.

The graph above displays the hazard function h(x). This equals f(x)/S(x), where f(x) is the pdf and S(x) = P(X > x) is the survival function.

The illustration above shows a point U chosen at random from a standard uniform distribution. The random variable X = 1/U has a standard Pareto distribution.

The simulation above shows a point U chosen at random from a standard uniform distribution on the y-axis. The light blue circle shows the value of the random variable X = 1/U on the x-axis, which has a standard Pareto distribution. The histogram accumulates the results of each simulation.

Y = xₘX1/α ∼ Pareto(α, xₘ) log(X − 1) ∼ Standard Logistic 1/X ∼ Standard Uniform

Chart of the related distribution Chart area for displaying the related pdf, cdf, and simulation

E(Y) = , Var(Y) =

Note that the parameter xₘ controls the scale of the distribution, while α controls the shape.

Note that:

  • A standard Pareto random variable X is the reciprocal 1/U of a standard uniform random variable U. So, log(X − 1) = log(1/U − 1) = log[(1 − U)/U].
  • This is the log odds of a standard uniform random variable, which is another way to derive a standard logistic distribution.

Note that the cdf of Y is FY(y) = P(Y ≤ y) = P(1/X ≤ y) = P(X ≥ 1/y) = y, from the cdf of the standard Pareto distribution. Hence, FY(y) = y, which is the cdf for a standard uniform random variable.