Fall 2011 Homework 6: Question 7
Let $X\sim N(\mu ,\sigma ^2)$ and $Y=e^{X}$. Then Y has a Log-Normal distribution (which means "log is Normal"; note that "log of a Normal" doesn't make sense since Normals can be negative). Find the mean and variance of Y using the MGF of X, without doing any integrals. Then for $\mu =0,\sigma =1$ find the nth moment $E(Y^{n})$ (in terms of n).
Solution: E(Y) = e^((u+o^2)/2). Var(Y) = e^(2u+2o^2). E(Y^n) = e^(n^2/2).
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."