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