What's the difference between the expectation of a function of a random variable and the law of the unconscious statistician

Question

Given a random variable $X$, some function $g(X)$, and $X$'s pdf $p_x(X)$ I know from probability that: $$\mathbb{E}(g(X)) = \int_x g(X)p_x(X) dx$$

In my reading, the Law of the Unconscious Statistician (LotUS) came up as a reason for one of steps of a proof in an academic paper. When I looked into the wiki link above, it seems to say the same thing as the equation above.

My question is, is there a difference between the two? Or is the LotUS just a formalism or a nickname for the expectation of a function of a random variable?

Answer 1

The "Law of the Unconscious Statistician" is just a name for the fact that $E(g(X))$ is given by the formula you wrote. There is no difference.

Answer 2

Finding $\mathbb Eg(X)$ can be done on several ways.

One of them is $$\mathbb Eg(X)=\int y\;dF_Y(y)$$where $Y:=g(X)$ and $F_Y$ denotes the CDF of $Y$.

Another (in almost all cases more convenient) is practicizing LOTUS:$$\mathbb Eg(X)=\int g(x)\;dF_X(x)$$where $F_X$ denotes the CDF of $X$.

This works if you know the distribution of $X$ and you can hold your shoulders for the distribution of $g(X)$.

In both cases a PDF or PMF might exist leading to variants.

One of them is mentioned by you:$$\mathbb Eg(X)=\int g(x)p_X(x)\;dx$$where $p_X$ denotes a PDF of $X$.