# Definition:Moment (Probability Theory)

## Definition

Let $X$ be a random variable on some probability space.

Let $a$ be a real number.

Then the $n$th moment of $X$ about $a$, usually denoted $\map {\mu_n} a$, is defined as:

$\map {\mu_n} a = \expect {\paren {X - a}^n}$

where $\expect X$ denotes the expectation of $X$.

### Discrete Random Variable

Let $X$ be a discrete random variable.

Then the $n$th moment of $X$ is denoted $\mu'_n$ and defined as:

$\mu'_n = \expect {X^n}$

where $\expect {\, \cdot \,}$ denotes the expectation function.