Find $Y = |X|$ for a PDF of X (Solution Check)

Question

I am given a PDF:

$$ f_X(x) = \frac{\ell}{2}e^{-\ell |x|} $$ with boundaries $-\infty < x < \infty$ and $\ell > 0$.

I am asked to find $f_Y(y)$ if $Y = |X|$.

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My try is: \begin{align} F_Y(y) &= P(Y \leq y)\\ &= P(-y \leq X \leq y)\\ &= F_X(y) - F_X(-y) \end{align}

Now I take the derivative of that, to find \begin{align} f_Y(y) &= f_X(y)\frac{dy}{dy} - f_X(-y)\frac{d(-y)}{dy}\\ &= f_X(y) + f_X(-y)\\ &= \frac{\ell}{2}e^{-\ell y} + \frac{\ell}{2}e^{-\ell y}\\ &= \ell e^{-\ell y} \end{align}

Is what I am doing correct?

Answer 1

Yes, that is correct. Also note that in the course of your calculations you effectively derived the general rule (for a continuous random variable):

$$f_{|X|}(x) = f_X(x) + f_X(-x).$$