How do you calculate the expected value of $E\left\{e^{-|X|}\right\}$ e.g. for Gaussian X?

Question

If $X$ is a random variable, I would like to be able to calculate something like $$E\left\{e^{-|X|}\right\}$$ How can I do this, e.g., for a normally distributed $X$?

Answer 1

If $X \sim N(0,1)$, and you have $Y=g(X) = e^{-|x|}$, so by definition $Y>0$. For $X<0, \ P(e^{X}<y) = P(X<\log y)$, because $e^X$ is an increasing function. For $X>0, P(e^{-X}<y) = P(-X>\log y) = P(X<-\log y),$ because $e^{-x}$ is a decreasing function. Putting together, $$ P(Y<y) = P(X<|\log y|) = \Phi(|\log y|), y>0 $$ Can you derive the pdf of $Y$ from here?

Answer 2

First of all observe that $e^X\sim$ lognormal

Second, observe that $-X$ has the same distribution of $X$ with opposite mean

Third: use the fact that lognormal is a known law (and thus with known mean) and the total probability theorem