Skewness of a Laplace variable

EasyRandom Variables Statistics~4m

Let $X$ have density $f(x) = \dfrac{\lambda}{2}e^{-\lambda|x - \mu|}$ for real $x$ (a Laplace distribution with $\lambda > 0$ ). Compute its skewness $\mathbb{E}\!\left[\left(\dfrac{X - \mu_X}{\sigma_X}\right)^{3}\right]$ .

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Is $f$ symmetric about $\mu$ ? What does that imply for odd central moments?
Symmetric $\Rightarrow$ mean $=\mu$ and all odd central moments $=0$ , so skewness $=0$ .

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Asked at: Citadel, Two Sigma

Skewness of a Laplace variable

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