I am faced with the following question:
Let X be a random variable with the binomial distribution B(n, p). Find the probability generating function F(t) for X. Using this generating function, compute the first three moments of X, that is, E[X], E[X^2], and E [X^3].
What tools do I need to do this?
Here is a selection of tools:
The probability mass function for $X\sim\mathcal{Bin}(n,p)$ is $$\mathsf P_X(x)=\binom n x p^x(1-p)^{n-x}$$
The expectation of a function $g$ of $X$ is $$\mathsf E(g(X)) = \sum_{x=0}^n g(x)\,\mathsf P_X(x)$$
The probability generating function is: $$\mathsf F_X(t)=\mathsf E(t^X)$$
More easily obtained by observing that for independent random variables $X_i$: $$\begin{align}\mathsf F_{\sum_{i=1}^nX_i}(t) ~=~& \prod_{i=1}^n\mathsf F_{X_i}(t)\end{align}$$
And when using the indicator random variables for success of the $i^\text{th}$ trial of the Binomial experiment: $$\begin{align}X~=~&\sum_{i=1}^n X_i\\\mathsf P_{X_i}(x)~=~&p\mathbf 1_{x=1}+(1-p)\mathbf 1_{x=0}\\\ \mathsf E(t^{X_i})~=~&(1-p)+tp\end{align}$$
The moment generating function, where $m_n$ is the $n^\text{th}$ moment, is $$\mathsf M_X(s) = \mathsf F_X(e^s) = 1+\sum_{n=1}^\infty \frac{s^n m_n}{n!}$$
Edit: and yes this is the long way around to finding the three moments.