Chapter 5.1, Problem 15E

If X is $N\left(\mu ,{\sigma }^2\right)$, then $M\left(t\right)=E\left(e^{tX}\right)=\exp \left(\mu t+\frac{{\sigma }^2{t}^2}{2}\right),-\infty <t<\infty$.

We then say that $Y=e^X$ has a lognormal distribution because $X=\ln Y$.

(a) Show that the pdf of Y is $g\left(y\right)=\frac{1}{y\sqrt{2\pi {\sigma }^2}}\exp \left[-\frac{{\left(\ln y-\mu \right)}^2}{2{\sigma }^2}\right],0<y<\infty$.

(b) Using $M\left(t\right)$, find (I) $E\left(Y\right)=E\left(e^X\right)=M\left(1\right)$.

(ii) $E\left(Y^2\right)=E\left(e^{2X}\right)=M\left(2\right)$, and (iii) $\mathrm{var}\left(Y\right)$.