Is |Z| a Positive Normal Distribution?

[Proggy99]

Homework Statement

Let $$\psi$$(x) = 2$$\phi$$(x) - 1. The function $$\psi$$ is called the positive normal distribution. Prove that if Z is standard normal, then |Z| is positive normal.

Homework Equations

The Attempt at a Solution

I am not really sure where to begin with this. Can anyone provide me a jumping off point, please?

I do know that $$\phi$$(-x) = 1 - $$\phi$$(x)
and so $$\phi$$(x) + $$\phi$$(-x) - 1 = 0.
I am not sure how to utilize that or if it is even on the right track. Thanks for any help.

 

[statdad]

Start with the cumulative distribution function of $$|Z|$$

$$P(|Z| \le x) = P(-x \le Z \le x) = \Phi(x) - \Phi(-x)$$

where $$\Phi$$ is the cdf of the standard normal. How can you simplify $$\Phi(-x)$$?

 

[Proggy99]

Ahhh, that makes perfect sense statdad. I kept trying to factor out 'x' when I looked at it the way you did it and got nowhere so discarded that method. I would substitute that with the equation I put in my first post to get the equation from the definition of positive normal. I just could not come up with that middle step to link the ideas until you put it so plainly. Thanks!

Start with the cumulative distribution function of $$|Z|$$

$$P(|Z| \le x) = P(-x \le Z \le x) = \Phi(x) - \Phi(-x)$$

where $$\Phi$$ is the cdf of the standard normal. How can you simplify $$\Phi(-x)$$?