Enter the scale parameter (x_m) and shape parameter (α), then provide a value x to compute P(X < x), P(X > x), and the probability density f(x) for a Pareto Type I distribution. You can also calculate a percentile by entering a probability and solving for x.
Probability
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Probability Density f(x)
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Cumulative Probability F(x)
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Survival Function P(X > x)
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Distribution Mean
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Distribution Variance
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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.
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.
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.
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.
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.
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.
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).
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.