The Folded Normal Distribution: PDF and Moments of |X|

Let $X \sim N(\mu, \sigma^2)$ with $\mu \geq 0$. Define $Y = |X|$.

(a) Derive the PDF of $Y$ for $y \geq 0$ by writing $P(Y \leq y) = P(-y \leq X \leq y)$ and differentiating.

(b) Show that when $\mu = 0$ the PDF simplifies to the half-normal distribution $f_Y(y) = \frac{\sqrt{2}}{\sigma\sqrt{\pi}}\, e^{-y^2/(2\sigma^2)}$ for $y \geq 0$.

(c) Derive $E[Y]$ and $\text{Var}(Y)$ for general $\mu, \sigma$. Express your answer using the standard normal PDF $\phi$ and CDF $\Phi$.

(d) Compute $E[Y]$ and $\text{Var}(Y)$ numerically for $\mu = 1$, $\sigma = 1$.