Power Series Expansion of Moment Generating Function

Question

Suppose that X ∼ Γ(2, λ). Use the power series expansion for $M_X (θ)$ to determine $E(X^4)$.

Working:

I know that $M_X(\theta)=(\frac{\lambda}{\lambda - \theta})^2 = \frac{\lambda ^2}{(\lambda - \theta)^2}$.

and I have calculated that

$E(X^4)= M_X^{(4)}(0)$ $=\frac{d^{(4)}}{d^{(4)}\theta} \frac{\lambda ^2}{(\lambda - \theta)^2}\Big|_{\lambda = 0} = \frac{5!}{\lambda^4} $

But I am unsure on how I would evaluate $E(X^4)$ using the power series expansion! Any help is appreciated!

Answer 1

Write $$\sum_{m=0}^\infty {\mathbb{E}(X^m)\,\theta^m\over m!}=M_X(\theta)={1\over (1-\theta/\lambda)^2}=\sum_{n=0}^\infty (n+1)\left({\theta\over\lambda}\right)^n. $$ Matching the coefficient of $\theta^4 $ gives $${\mathbb{E}(X^4)\over 4!}={5\over\lambda^4}. $$