The nth moment of a random variable X is
\mathbb{E}_{X} [X^{n}]
and the nth central moment of X is
\mathbb{E}_{X} [(X - \mathbb{E}_{X} [X])^{n}].
Definition 22.1 (Moment generating function) The moment generating function (MGF) of a random variable X is a function M_{X} (s) that is defined as
M_{X} (s) = \mathbb{E}_{X} [e^{s X}] = \sum_{x \in \mathbb{R}} e^{s x} \mathbb{P}_{X} (x).
Corollary 22.1 The nth moment of the random variable can be derived by taking the nth derivative of M_{X} (s) and evaluating it at s = 0
\mathbb{E}_{X} [X^{n}] = \frac{ d^{n} }{ ds^{n} } M_{X} (s) \left.\right|_{s = 0}.
Recall that the Taylor series for e^{x} is
e^{x} = 1 + x + \frac{x^{2}}{2!} + \dots = \sum_{i = 0}^{\infty} \frac{x^{i}}{i !}.
Therefore, the Taylor series of e^{s X} is
e^{s X} = \sum_{i = 0}^{\infty} \frac{(s X)^{i}}{i!} = \sum_{i = 0}^{\infty} \frac{s^{i} X^{i}}{i !}.
The moment generating function can be written as
M_{X} (s) = \mathbb{E}_{X} [e^{s X}] = \mathbb{E}_{X} \left[
\sum_{i = 0}^{\infty} \frac{s^{i} X^{i}}{i !}
\right] = \sum_{i = 0}^{\infty} \frac{\mathbb{E}_{X} [X^{i}] s^{i} }{i !}.
The nth moment is the coefficient of \frac{s^{k}}{k !} in the Taylor series ofM_{X} (s), which can be obtained by taking nth derivative M_{X} (s) and evaluating it at s = 0.
Properties of MGF
\begin{aligned}
M_{Y} (s)
& = \mathbb{E}_{X} \left[
e^{s (a X + b)}
\right]
\\
& = e^{b s} \mathbb{E}_{X} \left[
e^{s a X}
\right]
\\
& = e^{b s} M_{X} (a s).
\\
\end{aligned}
Corollary 22.3 Suppose X_{1}, \dots, X_{n} are independent random variables and X = \sum_{i = 1}^{n} X_{i}. Then we have the MGF of X as the sum of MGFs of X_{i}s,
M_{X} (s) = \prod_{i = 1}^{n} M_{X_{i}} (s).
\begin{aligned}
M_{X} (s)
& = \mathbb{E}_{X} [e^{s X}]
\\
& = \mathbb{E}_{X} [e^{s \sum_{i = 1} X_{i}}]
\\
& = \mathbb{E}_{X} [\prod_{i = 1}^{n} e^{s X_{i}}]
\\
& = \prod_{i = 1}^{n} [e^{s X_{i}}]
& [X_{i}\text{s are independent}]
\\
& = \prod_{i = 1}^{n} M_{X_{i}} (s)
\\
\end{aligned}