Question

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

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

where

$$\alpha ,\beta >0$$

. Find the density function.

Answer 1

Solution

$\text{Known:}$ A member of Pareto family of distributions has a distribution function given by:

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

Where $\alpha ,\beta >0$.

Recall the formula of density function:

$$f(y)=F^{\prime }(y)=\{\begin{array}{ll}\genfrac{}{}{0ex}{}{d}{dy}(0),& y<\beta \\ \genfrac{}{}{0ex}{}{d}{dy}\left[1-{\left(\genfrac{}{}{0ex}{}{\beta }{y}\right)}^{\alpha }\right],& y\ge \beta \end{array}$$

Hence,

$$f(y)=\{\begin{array}{ll}0,& y<\beta \\ \alpha {\beta }^{\alpha }{y}^{-\alpha -1},& y\ge \beta \end{array}$$

Therefore    f(y)={0,αβαy−α−1,​y<βy≥β​