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Pareto Distribution — Definition, Formula & Examples

The Pareto distribution is a continuous probability distribution that models phenomena where a large portion of outcomes come from a small fraction of causes — the classic "80/20 rule." It is defined on values greater than or equal to a minimum threshold and has a heavy right tail, meaning extreme values are more probable than in distributions like the normal.

A continuous random variable XX follows a Pareto distribution with shape parameter α>0\alpha > 0 and scale parameter xm>0x_m > 0 if its probability density function is f(x)=α xmαxα+1f(x) = \frac{\alpha\, x_m^{\alpha}}{x^{\alpha+1}} for x≥xmx \geq x_m, and f(x)=0f(x) = 0 otherwise. Its cumulative distribution function is F(x)=1−(xmx)αF(x) = 1 - \left(\frac{x_m}{x}\right)^{\alpha} for x≥xmx \geq x_m.

Key Formula

f(x)=α xmαxα+1,x≥xmf(x) = \frac{\alpha\, x_m^{\alpha}}{x^{\alpha+1}}, \quad x \geq x_m
Where:
  • α\alpha = Shape parameter (controls tail heaviness), must be positive
  • xmx_m = Scale parameter (minimum possible value of X), must be positive
  • xx = Value of the random variable

How It Works

The scale parameter xmx_m sets the minimum possible value of XX. The shape parameter α\alpha controls how rapidly the tail decays: smaller α\alpha produces a heavier tail with more extreme values. When α≤1\alpha \leq 1, the mean is infinite; when α≤2\alpha \leq 2, the variance is infinite. To find the probability that XX exceeds some value xx, use the survival function P(X>x)=(xmx)αP(X > x) = \left(\frac{x_m}{x}\right)^{\alpha}.

Worked Example

Problem: Incomes in a region follow a Pareto distribution with minimum income xm=30,000x_m = 30{,}000 and shape parameter α=3\alpha = 3. What is the probability that a randomly selected person earns more than $60,000?
Write the survival function: For a Pareto distribution, the probability of exceeding a value xx is:
P(X>x)=(xmx)αP(X > x) = \left(\frac{x_m}{x}\right)^{\alpha}
Substitute values: Plug in xm=30,000x_m = 30{,}000, x=60,000x = 60{,}000, and α=3\alpha = 3:
P(X>60,000)=(30,00060,000)3=(12)3P(X > 60{,}000) = \left(\frac{30{,}000}{60{,}000}\right)^{3} = \left(\frac{1}{2}\right)^{3}
Compute: Evaluate the expression:
P(X>60,000)=18=0.125P(X > 60{,}000) = \frac{1}{8} = 0.125
Answer: There is a 12.5% probability that a randomly selected person earns more than $60,000.

Why It Matters

The Pareto distribution appears in insurance (modeling large claims), network traffic analysis, and economics (wealth and income distributions). Understanding its heavy tail is essential in risk management, where underestimating the probability of extreme events can be costly.

Common Mistakes

Mistake: Using the Pareto PDF or CDF for values below the scale parameter xmx_m.
Correction: The distribution is only defined for x≥xmx \geq x_m. The probability density is exactly zero below this threshold — there is no probability mass to the left of xmx_m.