Name of the formula used to get the moments of a random vector

Question

I'm lloking for the name of the following formula used to get the moments of a random vector ($j \in \lbrace 1,...,n \rbrace$; $ k_1,...,k_j \in \lbrace 1,...,n \rbrace $; $r_1,...r_j \in \mathbb{N}$).

Answer 1

Contrary to popular belief, this is not a definition; it is the law of the unconscious statistician, with the expectation written in Riemann-Stieltjes form. There is a slight error in your formula, since you have too many differentials. What you should have is the following. Define the function $g$ by:

$$g(\mathbf{x}) = x_{k_j}^{r_1} \cdots x_{k_j}^{r_k}.$$

Then using the law of the unconscious statistician you have:

$$\begin{align} \mathbb{E}(g(\mathbf{X})) &= \int \cdots \int g(\mathbf{x}) \ dF_\mathbf{X}(\mathbf{x}) \\[6pt] &= \int \cdots \int x_{k_j}^{r_1} \cdots x_{k_j}^{r_k} \ dF_\mathbf{X}(\mathbf{x}). \\[6pt] \end{align}$$

Answer 2

Food for thought/discussion.

Let $X$ be a r.v. and $G=X^r$. We have the identity:

$E[G]=\int dF_{X}(x) x^r$ [1]

and:

$E[G]=\int dF_{G}(g) g$ [2]

Usually the equality between the two is referred as LOTUS, and comes from a change of variables.

But now what is the definition of $E[X^r]$? [1] or [2]? According to me [1] is the definition, because it is closer to what one would think from measure theory $\int_{\Omega} f(\omega) dP(\omega)$..

The "LOTUS theorem" than would be that we can evaluate [1] either from the definition, or from the image measure of $G$ (second formula), getting the same result.

Do you see it an other way?