Let X have the double exponential (or Laplace) distribution with $\alpha =0$, $\beta = 1$: $f_{X}(x)=\frac{1}{2}e^{-|x|}$, $-\infty < x < \infty$.
Show that $\varphi _{X}(u)=\frac{1}{1+u^{2}}$.
We are given a hint to use the result of the previous exercise (Let $\mu_{1},\cdots,\mu_{n}$ be probability measures. Suppose $\lambda_{j}\geq 0$ ($1\leq j \leq n$) and $\sum_{j=1}^{n}{\lambda_{j}}=1$. Let $\nu = \sum_{j=1}^{n}\lambda_{j}\mu_{j}$. Show that $\nu$ is a probability measure too, and that $\hat{\nu}(u)=\sum_{j=1}^{n}\lambda_{j}\hat{\mu_{j}}(u)$).
I am having trouble seeing how all of this information fits all together, and how the hint is supposed to help me. Am I supposed to break the density up into two densities because of the $|\, |$? And if so, how do I do this?
I thank you very much in advance.