Definition:Moment (Probability Theory)/Discrete

Definition

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.

That is:

$\mu'_n = \ds \sum_{x \mathop \in \Omega_X} x^n \map {p_X} x$

whenever this sum converges absolutely.

It can be seen from its definition that the expectation of a discrete random variable is its first moment.

Also see the relation between the variance of a discrete random variable and its second moment.

Sources

• 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $\S 4.3$: Moments