How do you find the Characteristic Function of a Standard Laplace Distribution:

Let X denote a real-valued random variable with an absolutely continuous distribution with density function p(x) = $\displaystyle \frac{1}{2}$$\displaystyle e^{-|x|}$, $\displaystyle \hspace{5mm}$-$\displaystyle \infty $< x < $\displaystyle \infty$.


Find the characteristic function.


Attempt:


I know that the setup should look something like this:


$\displaystyle \frac{1}{2}$$\displaystyle \int_{-\infty}^\infty$$\displaystyle e^{itx}$$\displaystyle e^{-|x|}$dx

Hey ishihara.

You should evaluate two integrals: one for (-infinity,0) and one for [0,infinity) by replacing |x| with -x on first integral and +x on second integral. Also recall that Integral e^(ax) = (1/a)*e^(ax) + C (indefinite integral that is).