Suppose $Y$ has a half-normal distribution with variance $\sigma^{2}=1$. Let $y_{p}$ be the $p^{t h}$ quantile of $Y$ such that $p=F\left(y_{p}\right)$ and $\Phi$ be the CDF of the standard normal distribution.

a. Show that $F\left(y_{p}\right)=2 \Phi\left(y_{p}\right)-1$.

b. Show that the $p^{t h}$ quantile of a half-normal is $y_{p}=\Phi^{-1}(p / 2+1 / 2)$.