Moment of a Random Variable

Suppose that $X$ is a discrete random variable with probability mass function $f(x)$ defined over $S$.

• $f(x)$ gives the probability of a value $x$

The $n$-th moment of $X$ is defined as:

$$E[X^n]=\sum _{x\in S}{x}^n\cdot f(x)$$

Thus, the expectation can be treated as the first moment.

Example:

$$\begin{array}{rl}E[X^2]& =\sum _{x\in S}{x}^2\cdot f(x)\\ & =1^2\cdot 0.25+2^2\cdot 0.5+3^2\cdot 0.25\\ & =4.5\end{array}$$