## moment generating function

The moment generating function of a normally distributed random variable is $M_{X}(t) = e^{\mu t + \sigma^{2} t^{2}/2}$. Find $E[X]$ and $\text{Var}(X)$.

So $M'_{X}(t) = (\mu + \sigma^{2}t) e^{\mu t + \sigma^{2} t^{2}/2}$ and thus $M'_{X}(0) = \mu$. Then find $E[X^2]$ by differentiating again and evaluating at $t = 0$, and use shortcut formula $\mbox{Var}(X) = E[X^2] - (E[X])^{2}$?

Couldn't you also use $R_{X}(t) = \ln(e^{\mu t + \sigma^{2} t^{2}/2})$, and the first and second derivatives would give you $E[X]$ and $\text{Var}(X)$ respectively?