13.4 Exercise 21

Let $M_{X} \left(\cdot\right)$ be the moment generating function for X.

1. Show that $Var \left[X\right]$ is the second derivative of $e^{- s \mu} M_{X} \left(s\right)$ evaluated at $s = 0$.
2. Use this fact to show that if $X \sim N \left(\mu , \sigma^{2}\right)$, then $Var \left[X\right] = \sigma^{2}$.