# What exactly falls under the term "moment" of a distribution?

Whenever people talk about the set of moments of a random variable X, they usually are referring to $\{E(X^n)|n\in \mathbb N \}$, where $E(X_n)$ is called the "$n$'th moment".

Also, $\{E[(X-E(X)]^n)|n\in \mathbb N \}$ is the set of "$n$'th central moments".

However, we can define a more general class, namely the set of functionals from probability distributions $P=\{F(\cdot)|F:\mathbb R\to [0,1],F \text { increasing}\}$ to real numbers:

$$M=\{J|J:P\to \mathbb R \}$$

For example, the Shannon information: $E[\ln(p(X))]\in M$.

Do we also refer to elements in this more general class $M$ as "moments" of the distribution of $X$? If not, what word do we use for it? More specifically, is there a named class of functionals of which both the mean, variance, etc, and the Shannon entropy are members?

• What kind of "functionals" are you referring to? That is, what properties do you suppose they have? Regardless, nobody understands "moment" in such a sense. In the most general sense, the word refers to expectations of powers of $|X|$.
– whuber
Jan 7 '18 at 15:29

In statistics and probability theory, a moment is the expected value of a power of the distance between a random variable and a point from the space where this variable takes values. For a random variable $\xi$ that takes values in a metric space, we can define a the moment of order $n$ about the point $c$, where $n$ is a non-negative integer writing:
$$m_n = E({ d(\xi,c)^n } ) \quad n \in N$$
If $\alpha$ is a real number, we can speak of a moment of order $\alpha$ writing:
$$m_{\alpha} = E( d(\xi,c)^{\alpha} )$$
For a real valued random variable $\xi$ and a non-negative integer $n$ we can define the moment of order $n$ about the point $c$ writing:
$$m_n = E((\xi - c)^n)$$