I am on page 163 of Degroot's probability and statistics (third edition), where he presents to me:
$g(y) = \frac{1}{2}(2-y)$ for $0<y<2$
He then mentions that the distribution function G of the given distribution is:
$G(y) = y - \frac{y^2}{4}$
Also, for $0<x<1$, the inverse function $y=G^{-1}(x)$ can be found by solving the equation $x=G(y)$ for y. The result is:
$$y=G^{-1}(x) = 2[1-(1-x)^{\frac{1}{2}}]$$
I cannot understand how he derives the values for $G(y)$ and $G^{-1}(x)$. I have only self-studied mathematics so this chapter has been difficult due to its wordyness and lack of examples.
$G(y)=\int\limits_0^yg(u)du$. $G^{-1}(x)$ is the result of solving the quadratic equation $y-\frac{y^2}{4}-x=0$ for $y$. General statement: $x=G(y)$ results in $y=G^{-1}(x)$