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

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.