Relationship between standard half-normal and standard normal distribution?

Question

Let X be a standard half-normal random variable with pdf:

$$f(x) = \cfrac{2}{\sqrt{2\pi}}e^{{-x^2}/{2}}$$

and let Z be a random variable such that P(Z = X) = 1/2 and P(Z = -X) = 1/2.

How to show that Z is a standard normal variable?

I tried the following:

$$P(Z<a) = P(Z<a|Z=X).P(Z=X) + P(Z<a|Z=-X).P(Z=-X)$$ $$\implies P(Z<a) = \cfrac{1}{2} [P(X<a) + P(X>-a)]$$

and then got stuck. Any pointers?

Answer 1

Note that if $a<0$, then we get, $$P(Z<a) = \dfrac12(0+P(X>-a)\\ =\int_{-a}^{\infty}\dfrac{1}{\sqrt{2\pi}}e^{-x^2/2}dx$$ Now, apply the substitution $t=-x$ to get, $$P(Z<a) = \int_{-\infty}^a\dfrac{1}{\sqrt{2\pi}}e^{-x^2/2}dx\\ =\Phi(a)$$ where $\Phi$ is the normal distribution function.

If $a\ge0$, then we get, $$P(Z<a) = \dfrac12(P(X<a)+1)\\ =\int_{0}^a\dfrac1{\sqrt{2\pi}}e^{-x^2/2}dx+\int_{-\infty}^0\dfrac1{\sqrt{2\pi}}e^{-x^2/2}dx\\ =\Phi(a)$$

Answer 2

Comment: Notice that for standard normal $X,$ you have $P(X < 1.5) + P(X > -1.5),$ etc.

In R: 'pnorm', with mean 0 and variance 1 by default, is the standard normal CDF. Then you have, for one value of $a:$

pnorm(1.5)
[1] 0.9331928
1 - pnorm(-1.5)
[1] 0.9331928