Chapter 5, Problem 132SE

To determine

Show that by expanding $e^{tX}$ in a power series and taking expectations term by term the moment generating function can be written as,

$M_X\left(t\right)=E\left(e^{tX}\right)=1+{{\mu }^{\prime }}_{1}t+{{\mu }^{\prime }}_2\frac{t^2}{2!}+\mathrm{...}+{{\mu }^{\prime }}_r\frac{t^r}{r!}+\mathrm{...}$.

Write the power series expansions for $M_X\left(t\right)$ for a gamma random variable and find ${{\mu }^{\prime }}_1$ and ${{\mu }^{\prime }}_2$.