Pareto distribution

Example

Let $x_m > 0$ and $\alpha > 0$. A random variable $X$ has the Pareto distribution with scale $x_m$ and shape $\alpha$ if $$P(X > x) = \left(\frac{x_m}{x}\right)^{\alpha} \quad \text{for } x \ge x_m, \qquad P(X > x) = 1 \quad \text{for } x < x_m.$$ This is a valid law: the distribution function $1 - (x_m/x)^\alpha$ is continuous and increasing from $0$ at $x_m$ to the limit $1$ as $x \to \infty$. Its density is $f(x) = \alpha x_m^{\alpha} / x^{\alpha + 1}$ for $x \ge x_m$ and $0$ otherwise. For $k > 0$, $E[X^k] = \int_{x_m}^\infty \alpha x_m^\alpha x^{k - \alpha - 1} \, dx$ is finite if and only if $k < \alpha$, and then equals $\frac{\alpha x_m^k}{\alpha - k}$. So the mean is $\frac{\alpha x_m}{\alpha - 1}$ for $\alpha > 1$ and infinite for $\alpha \le 1$, and the variance is $\frac{\alpha x_m^2}{(\alpha - 1)^2 (\alpha - 2)}$ for $\alpha > 2$.