How to show that $Z = |X|$ and $-|X|$ has a a standard normal distribution? ($X \sim$ Folded standard normal distributed)

Question

Given the probability density function of $X$ (folded standard normal distributed) is:

$$f(x) = \frac{2}{\sqrt{2 \pi}} \exp\left(-\frac{x^2}{2}\right),\quad x \geqslant 0 $$

How can one show that $Z = |X|$ with probability $1/2$ and $Z=-|X|$ with probability $1/2$ has a standard normal distribution?

Please give me a hint only. Thanks!

(I tried to show that E(Z) = 0 and Var(Z) = 1, but it was not sufficient.)

Answer 1

Hints

• Since the density of $X$ has zero mass on the negative real line, we have, with probability 1, that $X \ge 0$. So you can discard the absolute value sign around $X$. That is, $Z = X$ or $-X$ with equal probability. More precisely, there is a symmetric Bernoulli variable $$ B \sim \begin{cases} 0 & \text{with probability}\; 1/2 \\ 1 & \text{with probability}\; 1/2 \end{cases} $$ which is independent of $X$ and $Z = B X + (1-B)(-X)$. (This a more detailed description of for how $Z$ is constructed from $X$.)

• You can now use the law of total probability. For any (measurable!) subset of the real-line (say $A=(-\infty,x]$), we have $$ P (Z \in A) = P(X \in A|B = 0) P(B = 0) + P(-X \in A| B = 1) P(B = 1) $$ You should be able to figure out the rest.