Pareto Distribution Calculator

Compute the Pareto distribution PDF, CDF, upper tail probability, mean and variance from scale xm and shape alpha. Visualize the distribution behind the 80-20 rule with charts.

Input

For a Pareto distribution with scale xm and shape alpha, compute the probability density, cumulative distribution, upper tail, mean and variance at a given x. This is the distribution behind the 80-20 rule.

Value of x

The point to evaluate. It is meaningful for x at least the scale xm.

Scale xm (minimum, positive)

The smallest value the distribution can take. Enter a positive number.

Shape alpha (positive)

The parameter controlling tail heaviness. Smaller values give heavier tails.

Result

Probability that x is at most 2 (cumulative distribution CDF)

0.75

Upper tail Q(x)

0.25

Density f(x)

0.25

Mean

Variance

Diverges (not finite)

Probability density PDF

Cumulative distribution CDF

How it works

The probability density is f(x) = alpha xm^alpha / x^(alpha+1) for x at least xm, and 0 when x is below xm.
The cumulative distribution (lower probability) is F(x) = 1 minus (xm / x)^alpha, and the upper tail probability is (xm / x)^alpha.
The mean is finite only when shape alpha is greater than 1, equal to alpha xm / (alpha minus 1), and diverges when alpha is at most 1.
The variance is finite only when shape alpha is greater than 2, equal to xm squared times alpha divided by ((alpha minus 1) squared times (alpha minus 2)).
Both the scale xm and the shape alpha must be positive values.
A smaller shape alpha gives a heavier tail, strengthening imbalances like the 80-20 rule.

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