# question about a "half normal" distribution? Thanks for checking this out.

• Nov 19th 2009, 04:40 AM
Intsecxtanx
question about a "half normal" distribution? Thanks for checking this out.
Let X be N (0, 1)

(a) How do you find the p.d.f of |X|, a distribution that is often called the half normal?

(b) How do you find the expectation of |X| ?

Thank you for your time. (Happy)
• Nov 19th 2009, 11:47 AM
Focus
Quote:

Originally Posted by Intsecxtanx
Let X be N (0, 1)

(a) How do you find the p.d.f of |X|, a distribution that is often called the half normal?

(b) How do you find the expectation of |X| ?

Thank you for your time. (Happy)

Hint: \$\displaystyle \mathbb{P}(|X| =a)=\mathbb{P}(X=a)+\mathbb{P}(X=-a)\$ if \$\displaystyle a\geq 0\$ and zero otherwise.

Second hint: Normals are symmetric about their mean.

After you find the pdf integrate to find the expectation. If you need help with that, post here.
• Nov 19th 2009, 04:38 PM
theodds
Well, \$\displaystyle P(|X| = a)\$ is zero for all a. But,\$\displaystyle F_{|X|} (x) = P(|X| \le x) = P(-x \le X \le x) = F_X (x) - F_X (-x)\$, which is enough to get the pdf by taking the derivative WRT x (note that, up to this point and after taking the derivative, we haven't used any information about the distribution of X). To find the expectation, just for simplicity let's set Y = |X| then find the expectation of Y in the usual way, now that we have the pdf for it. The function you need to integrate has an easy enough antiderivative.
• Nov 19th 2009, 05:37 PM
davismj