Chapter 6.3, Problem 18E

A member of the Pareto family of distributions (often used in economics to model income distributions) has a distribution function given by

$F(y)=\{\begin{array}{ll}0,\hfill & y<\beta ,\hfill \\ 1-{\left(\frac{\beta }{y}\right)}^a,\hfill & y\ge \beta ,\hfill \end{array}$

where α, β > 0.

a Find the density function.

b For fixed values of β and α, find a transformation G(U) so that G(U) has a distribution function of F when U has a uniform distribution on the interval (0, 1).

c Given that a random sample of size 5 from a uniform distribution on the interval (0, 1) yielded the values .0058, .2048, .7692, .2475 and .6078, use the transformation derived in part (b) to give values associated with a random variable with a Pareto distribution with α = 2, β = 3.