Chapter 2.5, Problem 5E

Let the moment-generating function $M\left(t\right)$ of X exist for $-h<t<h$.

Consider the function $R\left(t\right)=\ln M\left(t\right)$.

The first two derivatives of $R\left(t\right)$ are, respectively $R\text{'}\left(t\right)=\frac{M\text{'}\left(t\right)}{M\left(t\right)}$ and $R"\left(t\right)=\frac{M\text{'}\left(t\right)M"\left(t\right)-{\left[M\text{'}\left(t\right)\right]}^2}{{\left[M\left(t\right)\right]}^2}$.

Setting $t=0$, show that

(a) $\mu =R\text{'}\left(0\right)$.

(b) ${\sigma }^2=R"\left(0\right)$.