A random variable $X$ is lognormal if ln $X$, the natural logarithm of $X$, is normally distributed. Find the mean and variance of a lognormal random variable with $ln X ∼ N(µ, σ^2)$
How should I solve this using the moment generating function of normal distribution, without differentiation?
You can write $X=e^{\mu+\sigma U}$ where $U$ has standard normal distribution.
Then $\mathbb EX^n=\mathbb Ee^{n\mu+n\sigma U}=e^{n\mu}M_U(n\sigma)$ where $M_U$ denotes the MGF of $U$.
Apply this for $n=1,2$ and find the variance as $\mathbb EX^2-(\mathbb EX)^2$.
Write $Y=\ln X$ so $X^t=e^{tY}$. You get the mean of powers of $X$ from the mgf of $Y$.In particular only the mgf is needed, not its derivatives.
