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- Virtual Laboratories
- 4. Special Distributions
- The Pareto Distribution

The Pareto distribution is a skewed, heavy-tailed distribution that is sometimes used to model the distribution of incomes and other financial variables.

Let \(a \gt 0\) be a parameter. The function \(F\) given below is a distribution function.

\[ F(x) = 1 - \frac{1}{x^a}, \quad 1 \le x \lt \infty \]The distribution defined by the function in Exercise 1 is called the Pareto distribution with shape parameter \(a\), and is named for the economist Vilfredo Pareto.

The probability density function \(f\) is given by

\[ f(x) = \frac{a}{x^{a+1}}, \quad 1 \le x \lt \infty \]The probability density function \(f\) satisfies the following properties:

- \(f\) is decreasing.
- The mode occurs at \(x = 1\).
- \(f(x) \to 0\) as \(x \to \infty\)

The reason that the Pareto distribution is heavy-tailed is that the convergence in part (c) is at a *power rate* rather than an *exponential rate*.

In the simulation of the special distribution simulator, select the Pareto distribution. Vary the shape parameter and note the shape and location of the density function. For selected values of the parameter, run the simulation 1000 times and note the apparent convergence of the empirical density to the true density.

The quantile function is

\[ F^{-1}(p) = \frac{1}{(1 - p)^{1/a}}, \quad 0 \le p \lt 1 \]Find the median and the first and third quartiles for the Pareto distribution with shape parameter \(a = 3\). Compute the interquartile range.

In the special distribution calculator, select the Pareto distribution. Vary the shape parameter and note the shape and location of the density function and the distribution function.

Because the Pareto distribution is heavy-tailed, the mean, variance, and other moments are finite only if the shape parameter \(a\) is sufficiently large.

Suppose that \(X\) has the Pareto distribution with shape parameter \(a \gt 0\). Then

- \(\E(X^n) = \frac{a}{a - n}\) if \(0 \lt n \lt a\)
- \(\E(X^n) = \infty\) if \(n \ge a\)

In particular, the mean and variance of \(X\) are

- \(\E(X) = \frac{a}{a - 1}\) if \(a \gt 1\)
- \(\var(X) = \frac{a}{(a - 1)^2 (a - 2)}\) if \(a \gt 2\)

In the special distribution simulator, select the Pareto distribution. Vary the parameters and note the shape and location of the mean/standard deviation bar. For each of the following parameter values, run the simulation 1000 times and note the behavior of the empirical moments:

- \(a = 1\)
- \(a = 2\)
- \(a = 3\)

As with many other distributions, the Pareto distribution is often generalized by adding a scale parameter. Thus, suppose that \(Z\) has the basic Pareto distribution with shape parameter \(a \gt 0\). If \(b \gt 0\), the random variable \(X = b \, Z\) has the Pareto distribution with shape parameter \(a\) and scale parameter \(b\). Note that \(X\) takes values in the interval \([b, \infty)\).

Analogies of the results given above follow easily from basic properties of the scale transformation.

The probability density function is

\[ f(x) = \frac{a \, b^a}{x^{a + 1}}, \quad b \le x \lt \infty \]The distribution function is

\[ F(x) = 1 - \left( \frac{b}{x} \right)^a, \quad b \le x \lt \infty \]The quantile function is

\[ F^{-1}(p) = \frac{b}{(1 - p)^{1/a}}, \quad 0 \le p \lt 1 \]The moments are given by

- \(\E(X^n) = b^n \, \frac{a}{a - n}\) if \(0 \lt n \lt a\)
- \(\E(X^n) = \infty\) if \(n \ge a\)

The mean and variance are

- \(\E(X) = b \, \frac{a}{a - 1}\) if \(a \gt 1\)
- \(\var(X) = b^2 \, \frac{a}{(a - 1)^2 (a - 2)}\) if \(a \gt 2\)

Suppose that the income of a certain population has the Pareto distribution with shape parameter 3 and scale parameter 1000. Find each of the following:

- The proportion of the population with incomes between 2000 and 4000.
- The median income.
- The first and third quartiles and the interquartile range.
- The mean income.
- The standard deviation of income.
- The 90th percentile.

- \(\P(2000 \lt X \lt 4000) = 0.1637\) so the proportion is 16.37%
- \(Q_2 = 1259.92\)
- \(Q_1 = 1100.64\), \(Q_3 = 1587.40\), \(Q_3 - Q_1 = 486.76\)
- \(\E(X) = 1500\)
- \(\sd(X) = 866.03\)
- \(F^{-1}(0.9) = 2154.43\)

The following exercise is a restatement of the fact that \(b\) is a scale parameter.

Suppose that \(X\) has the Pareto distribution with shape parameter \(a\) and scale parameter \(b\). If \(c \gt 0\) then \(c \, X\) has the Pareto distribution with shape parameter \(a\) and scale parameter \(b \, c\).

If \(X\) has the basic Pareto distribution with shape parameter \(a\) then \(1 / X\) has the beta distribution with left parameter \(a\) and right parameter 1.