Moments of Discrete Random Variables

Let $X$ be a discrete random variable with PMF $p(x)$. For a positive integer $k$, the $k$-th moment of $X$ is defined as

$$E[X^k] \;=\; \sum_x x^k\, p(x).$$

This is the expectation of the transformed variable $X^k$ -- a direct application of $E[g(X)] = \sum_x g(x)\,p(x)$ with $g(x)=x^k$. The first moment $E[X]$ is the mean of $X$. The second moment $E[X^2]$ measures the average squared value of $X$ and appears throughout the theory of variance.

For example, let $X$ have PMF

$$P(X=0)=\dfrac{1}{4},\qquad P(X=1)=\dfrac{1}{2},\qquad P(X=2)=\dfrac{1}{4}.$$

Then

$$\begin{align*} E[X^2] &= 0^2\cdot\dfrac{1}{4} \;+\; 1^2\cdot\dfrac{1}{2} \;+\; 2^2\cdot\dfrac{1}{4} \\[3pt] &= 0 + \dfrac{1}{2} + 1 \\[3pt] &= \dfrac{3}{2}. \end{align*}$$

The same recipe gives $E[X^3]$, $E[X^4]$, and so on -- just replace the exponent.