1. ## random variable function

If $\displaystyle X \sim N( \mu , {\sigma}^2)$, then how do you derive the density of $\displaystyle Y = |X|$?
This is what I did:

$\displaystyle F_{y}(y) = P(Y < y)$
$\displaystyle F_{y}(y) = P(|X| < y)$
$\displaystyle F_{y}(y) = \left\{\begin{array}{cc}P(X< y),&\mbox{ if } x > 0 \\ P(-X <y), & \mbox{ if } x<0\end{array}\right.$

Is this right so far?

2. The last line is off, you don't want to change the RV X, you want to play with the constant y.

$\displaystyle F_Y(y) = P(Y < y) = P(|X| < y)$

$\displaystyle = P(-y<X < y)=F_X(y)-F_X(-y)$

You can now differentiate wrt y, and obtain

$\displaystyle f_Y(y) =f_X(y)+f_X(-y)$

3. I got
$\displaystyle \frac{exp(\frac{-y^2}{2 \sigma^2})}{\sigma \sqrt{2 \pi}}$
I'm pretty sure this is right, but can someone vouch for me? Thanks.

4. Did you really add the two densities?
AND do note the domain of Y.

5. Ahh, there's a typo on my original question. If $\displaystyle \mu =0$, then am I right?
y>0

6. Originally Posted by ezong
Ahh, there's a typo on my original question. If $\displaystyle \mu =0$, then am I right?
y>0