Moment generating function for $f(x) = \frac{1}{2}e^{-|x|}$

Question

I am supposed to calculate the first expectation value from the prob.den.function above. So I know that you need to calculate the expectation value of $e^{tx}$. To do that I have chopped the integral to two pieces $$\frac{1}{2}\int_{-\infty}^{+\infty} e^{tx}e^{-|x|} dx = \frac{1}{2}\int_{-\infty}^{0} e^{tx^2}dx + \frac{1}{2}\int_{0}^{\infty} e^{-tx^2}dx $$ and this becomes $$ \frac{\sqrt{\pi}}{4}\left(\frac{1}{\sqrt{-t}}+ \frac{1}{\sqrt{t}}\right)$$ This is clearly not defined at $t = 0$, so have I made mistake or is there something that I dont get? Thanks!

Answer 1

The mistake you made was in simplifying your exponentials.

$$ e^{tx}e^x = e^{tx+x} = e^{(t+1)x}$$

not $e^{tx^2}.$ Likewise for the other pair.