Pareto Distribution Calculator

Enter Calculation

Function selection Probability density f(x, xm, alpha) Lower cumulative distribution P(x, xm, alpha) Upper cumulative distribution Q(x, xm, alpha)

Scale parameter xm

Shape parameter alpha

Initial value of x

Increment

Number of repetitions

Results

Selected function value

1

evaluated at x

Probability density f	1
Lower cumulative probability P	0
Upper cumulative probability Q	1

What is the Pareto Distribution Calculator?

This tool evaluates the Pareto (Type I) distribution — a classic heavy-tailed power-law distribution used to model wealth, city sizes, file sizes and the "80/20" rule. Given a scale parameter xm (the minimum value) and a shape parameter alpha, it computes the probability density f, the lower cumulative probability (CDF) P, and the upper cumulative probability (survival function) Q at any point x. It is a pure-math statistics tool with no regional assumptions.

Pareto PDF \(f(x)\): the curve starts at the scale \(x_m\) and decays with a heavy tail, steeper for larger \(\alpha\).

How to use it

Pick which function you want (density, CDF, or survival). Enter the scale parameter xm (must be greater than 0), the shape parameter alpha (must be greater than 0), and the point x (the initial value of x). The optional increment and number of repetitions describe a series of x points for plotting a graph: \(x_i = \text{startX} + i \times \text{stepX}\) for \(i = 0 \ldots \text{loopCount} - 1\).

The formula explained

For \(x \ge x_m > 0\) and \(\alpha > 0\):

Density: $$f(x) = \frac{\alpha\;x_m^{\alpha}}{x^{\,\alpha+1}}$$ CDF: $$P(x) = 1 - \left(\frac{x_m}{x}\right)^{\alpha}$$ Survival: $$Q(x) = \left(\frac{x_m}{x}\right)^{\alpha}$$ Note that \(P + Q = 1\) always. When \(x < x_m\) the support is not yet reached, so \(f = 0\), \(P = 0\) and \(Q = 1\).

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CDF \(P(x)\) and survival \(Q(x)\) are complementary: \(P(x) + Q(x) = 1\).

Worked example

Let \(x_m = 1\), \(\alpha = 1\), \(x = 2\). Then $$f = \frac{1 \cdot 1}{2^{2}} = 0.25,$$ $$P = 1 - \left(\frac{1}{2}\right)^{1} = 0.5,$$ and $$Q = \left(\frac{1}{2}\right)^{1} = 0.5.$$ Check: \(P + Q = 1.0\). For \(x_m = 2\), \(\alpha = 3\), \(x = 4\): $$f = \frac{3 \cdot 8}{256} = 0.09375,$$ $$P = 1 - 0.125 = 0.875,$$ \(Q = 0.125\).

FAQ

What does the shape parameter alpha control? Larger \(\alpha\) makes the tail decay faster (less heavy); the mean is finite only when \(\alpha > 1\).

Why is the density zero below xm? The Pareto distribution has support only on \(x \ge x_m\), so no probability mass exists below the scale parameter.

What is the survival function? \(Q(x)\) is the probability that a random variable exceeds \(x\), i.e. the complementary CDF, equal to \(1 - P(x)\).