Definition:Moment (Probability Theory)

This page is about moment in the context of probability theory. For other uses, see moment.

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.

Also see

• Definition:Raw Moment
• Definition:Central Moment
• Definition:Standardized Moment

• Results about moments in the context of probability theory can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): moment
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): moment
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): moment (in statistics)

• Weisstein, Eric W. "Moment." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Moment.html