Moment generating function $Y$

Question

If $X$ is a random variable, normally distributed with unknown parameters how could I find the mgf of random variable $Y$, where Y=$e^X$?

I am able to find mgf of $X$ from the mgf of a standard normally distributed random variable but not for $Y$. Any tips would be appreciated.

Thanks

Answer 1

The moment generating function of $Y$ is infinity at all points except $0$. In fact $Ee^{te^{X}} \geq P\{X\geq c\} e^{te^{c}}\to \infty $ as $t \to \infty$ because $P\{X\geq c\}\geq \alpha c^{-1}e^{-c^{2}/2}$ for some positive constant $\alpha$.