Sufficient statistic $\sum_{j=1}^{n} |x_{j}|$ for laplace distribution Let be $X_{1},\ldots , X_{n}$ random variables independent and identically distributed with density function:
$$ f_{\theta}(x)=\dfrac{1}{2}e^{-|x-\mu|}, \quad x,\mu \in \mathbb{R} $$
Find the joint density function $X=(X_{1},\ldots X_{n})$ and show that $\sum_{j=1}^{n} |x_{j}|$ is a sufficient statistic for $\mu$.
I have that, $f_{\theta}(x_{1},\ldots,x_{n})=\dfrac{1}{2^{n}}e^{-\sum_{i=1}^{n}|x_{i}-\mu|}$ but I do not know how to use the factorization theorem to show that  $\sum_{j=1}^{n} |x_{j}|$ is a sufficient statistic for $\mu$.
Also, How can I express this density function as an exponential family? I have problems with the expression $e^{-\sum_{i=1}^{n}|x_{i}-\mu|}$.


I do not know how to use the factorization theorem

You have to write the likelihood as a product of two functions. One function is allowed to have parameters in it, and one is not. The one with the parameter in it will have the sufficient statistic in it.

Also, How can I express this density function as an exponential
family? I have problems with the expression

Once you have a definition of an example family (sometimes they are written in different ways), you just need to write your likelihood like that. It will simply be notation matching.
