# Distribution of Sum of Absolute Value of Random Variable

I'm wondering if there's a distribution $\Gamma$ such that if we draw $x_i \dots d_n$ iid from $\Gamma$, then $\sum_{i=1}^n |x_i|$ has a nice distribution. I thought maybe the normal distribution, since then the absolute value would be half-normal, but I'm not sure what the distribution of the sum of half-normal variables are.

• If you're willing to consider a distribution that's supported on the positive reals to begin with, there's a nice expression for the distribution of a sum of independent exponential random variables (en.wikipedia.org/wiki/Erlang_distribution) – Julian Rosen Sep 20 '12 at 4:18

If i.i.d. random variables $X_1, X_2, \ldots, X_n$ have Laplacian density $$f(x) = \frac{1}{2}\exp\left(-|x|\right), -\infty < x < \infty,$$ then $|X_i|$ are i.i.d. exponential random variables with mean $1$, and $\sum_{i=1}^n |X_i|$ is a Gamma random variable with mean $n$ and order parameter $n$. This is effectively the same result as the one in the comment by @PinkElephants but without the requirement of support on the positive reals only (which makes the distinction between $X_i$ and $|X_i|$ irrelevant.)