Pareto distribution

A continuous probability distribution with density

$$ p( x) = \left \{ \begin{array}{ll} \frac \alpha {x _ {0} } \left ( \frac{x _ {0} }{x} \right ) ^ {\alpha + 1 } , & x _ {0} < x < \infty , \\ 0, & x \leq x _ {0} , \\ \end{array} $$

depending on two parameters $ x _ {0} > 0 $ and $ \alpha > 0 $. As a "cut-off" version the Pareto distribution can be considered as belonging to the family of beta-distributions (cf. Beta-distribution) of the second kind with the density

$$ \frac{1}{B( \mu , \alpha ) } \frac{x ^ {\mu - 1 } }{( 1+ x) ^ {\mu + \alpha } } ,\ \ \mu , \alpha > 0,\ \ 0 < x < \infty , $$

for $ \mu = 1 $. For any fixed $ x _ {0} $, the Pareto distribution reduces by the transformation $ x = x _ {0} /y $ to a beta-distribution of the first kind. In the system of Pearson curves the Pareto distribution belongs to those of "type VI" and "type XI" . The mathematical expectation of the Pareto distribution is finite for $ \alpha > 1 $ and equal to $ \alpha x _ {0} /( \alpha - 1) $; the variance is finite for $ \alpha > 2 $ and equal to $ \alpha x _ {0} ^ {2} /( \alpha - 1) ^ {2} ( \alpha - 2) $; the median is $ 2 ^ {1/ \alpha } x _ {0} $. The Pareto distribution function is defined by the formula

$$ {\mathsf P} \{ X < x \} = 1 - \left ( \frac{x _ {0} }{x} \right ) ^ \alpha ,\ \ x > x _ {0} ,\ \ \alpha > 0. $$

The Pareto distribution has been widely used in various problems of economical statistics, beginning with the work of W. Pareto (1882) on the distribution of profits. It is sometimes accepted that the Pareto distribution describes fairly well the distribution of profits exceeding a certain level in the sense that it must have a tail of order $ 1/x ^ \alpha $ as $ x \rightarrow \infty $.

References

[1]	H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946)

References

[a1]	N.L. Johnson, S. Kotz, "Distributions in statistics: continuous univariate distributions" , Houghton Mifflin (1970)
[a2]	H.T. Davis, "Elements of statistics with application to economic data" , Amer. Math. Soc. (1972)