Chapter 8, Problem 8.16TE

(a)

To determine

To show:

  $L\left(p\right)=\frac{1}{\mu }{\displaystyle {\int }_0^p{F}^{-1}\left(y\right)dy}$ Of the given density function.

(b)

To determine

To show:

  $L\left(p\right)$ Is convex set of the given density function.

(c)

To determine

To show:

  ${\displaystyle \underset{0}{\overset{1}{\int }}L\left(p\right)dp}=\frac{1}{\mu }{\displaystyle \underset{0}{\overset{\infty }{\int }}\left(1-F\left(x\right)\right)xf\left(x\right)dx}$ Of the given density function.

(d)

To determine

To show:

The Gini index of a uniform $\left(0,1\right)$ and an exponential random variable of the given density function.