$X$ is a random variable for normal distribution: $X\sim N(\mu, \sigma^2)$.
What is the mean and variance of $e^{X}$?
My attempt:
$$E[e^{X}]=e^{E[x]} \text{, by the invariance property?}$$
$$\operatorname{var}(e^{x})=e^{\operatorname{var}(x)}, \text{ similarly}$$
This looks too easy, probably not right.
Should I look at $e^{X}$ as a whole. use moment generating function?
But normal pdf requires $e^{x^2}$. I'm stuck.