Chapter 8, Problem 8.28P

Suppose that $L\left(p\right)$ is the Lorenz curve associated with the random variable X and that $c>0$.

a. Find the Lorenz curve associated with the random variable cX.

b. Show that $L\left(p\right)$, the Lorenz curve associated with the random variable $X+c$, is $L_c\left(p\right)=\frac{L\left(p\right)E\left[X\right]+pc}{E\left[X\right]+c}$

c. Verify that the answer to part (b) is in accordance with the formulas given in Example 7a in the case that X is uniform over the interval $\left(0,b-a\right)$ and $c=a$.