Density of $Y=|X|$, where $X\sim N(0,\sigma^2)$

Question

I am reviewing for an upcoming exam, and came across this question in the textbook. Can someone please help me with this question? Thanks.

If $X\sim N(0,\sigma^2)$, then find the density of $Y=|X|$.

Answer 1

If you're having trouble finding distributions (or densities), it's usually not a bad idea to start from the cdf, $F_X(x) = P(X \leq x)$.

Let $Y = |X|$, $X \sim N(0,\sigma^2)$ and let $F_X$ be the cdf of $X$. Then, for $y \geq 0$, \begin{align*} P(Y \leq y) &= P(|X| \leq y) \\ &= P(-y \leq X \leq y) \\ &= P(-y \leq X \leq 0) + P(0 < X \leq y) \tag{1}\\ &= P(0 \leq X \leq y) + P(0 < X < \leq y) \tag{2} \\ &= 2P(0\leq X \leq y) \tag{3} \\ &= 2(F_X(y) - F_X(0)). \end{align*} In (1) we were able to split the union of events into a sum, since they are mutually exclusive. In (2) we used symmetry of the centred normal random variable $X$. In (3) we used continuity of $X$ to change the $<$ to a $\leq$.

To conclude you need only use the relationship between $F_Y$ and its density $f_y$.