What is the Pareto distribution?
The Pareto distribution is a power-law probability distribution used to model phenomena where a small proportion of causes account for a large proportion of effects — the so-called 80/20 rule. It is defined by two parameters: scale (xm) and shape (α), and is commonly used in economics, insurance, and reliability analysis. See also our calculate SMp(x) at given x, Approximate Mean & Approximate Variance — SMp(x) Distribution.
What are the scale (xm) and shape (α) parameters?
The scale parameter xm represents the minimum possible value of the random variable — no observation can be less than xm. The shape parameter α (alpha) controls the heaviness of the tail: a smaller α produces a heavier tail (more extreme values), while a larger α creates a thinner, lighter tail.
What is the formula for the Pareto PDF?
The probability density function is f(x) = (α · xm^α) / x^(α+1) for x ≥ xm, and f(x) = 0 for x < xm. It describes the relative likelihood of a random variable taking a given value.
How is the Pareto CDF calculated?
The cumulative distribution function is F(x) = 1 − (xm / x)^α for x ≥ xm. It gives the probability that the random variable X is less than or equal to x. For x < xm, F(x) = 0. You might also find our t-Distribution Calculator useful.
What does P(X > x) mean in Pareto distribution?
P(X > x) is the survival function, also called the right-tail probability. It equals (xm / x)^α for x ≥ xm, and represents the probability that the random variable exceeds the value x. This is especially useful in risk and insurance modeling.
When does the Pareto distribution have a finite mean and variance?
The mean of the Pareto distribution is finite only when α > 1, and equals α·xm / (α − 1). The variance is finite only when α > 2, and equals xm² · α / ((α−1)² · (α−2)). For α ≤ 1 the mean is infinite, and for α ≤ 2 the variance is infinite.
How do I find a Pareto percentile?
To find the x value corresponding to a given probability p (the percentile), use the inverse CDF formula: x = xm / (1 − p)^(1/α). For example, the 80th percentile (p = 0.80) gives x = xm / (0.20)^(1/α).
What is the difference between Pareto Type I and other Pareto types?
Pareto Type I is the classical form with support x ≥ xm, defined by scale and shape parameters. Pareto Type II (Lomax distribution) shifts the support to x ≥ 0 by replacing x with (x + xm). Higher types add additional location and shape parameters. This calculator uses the Pareto Type I distribution.