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.
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.
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.
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.