Finding the Moment Generating Function given f(x)

Question

I think I'm having trouble with this because of the absolute value. Otherwise, I know how to solve for the moment generating function. The problem is that show that if a random variable has the probability density

$$ f(x) = \frac12 e^{-|x|} \qquad -\infty\lt X \lt \infty $$

its moment-generating function is given by

$$ M_x(t) = \frac1{1-t^2} $$

Answer 1

The moment generating function is defined $M_X(t) = \mathbb{E}[e^{tX}]$. In your case this is

$\int_{-\infty}^\infty e^{tx} \frac{1}{2}e^{-|x|} dx$ =

$\frac{1}{2}\int_{-\infty}^\infty e^{tx-|x|} dx$ =

$\frac{1}{2}(\int_{-\infty}^0 e^{tx-|x|} dx + \int_{0}^\infty e^{tx-|x|} dx)$ =

$\frac{1}{2}(\int_{-\infty}^0 e^{tx+x} dx + \int_{0}^\infty e^{tx-x} dx)$

You shouldn't have trouble evaluating those integrals now.