Exponential of a Normal: The Log-Normal Distribution

Let $X \sim N(\mu, \sigma^2)$ and define $Y = e^X$.

(a) Derive the PDF of $Y$ using the change-of-variables formula.

(b) Using the MGF of the normal distribution, compute $E[Y]$ and $\operatorname{Var}(Y)$.

(c) Show that the median of $Y$ is $e^{\mu}$ and explain why $E[Y] > \text{median}(Y)$ when $\sigma > 0$.