Law of the Unconscious Statistician

Computing expectations of functions of random variables.

The Law of the Unconscious Statistician (LOTUS) says you don't need to find the distribution of $g(X)$ to compute its expectation. You can integrate $g$ against the density of $X$ directly:

E[g(X)] = \int g(x)\, f_X(x)\, dx \quad \text{(continuous)}

E[g(X)] = \sum_x g(x)\, P(X = x) \quad \text{(discrete)}

The "unconscious" name comes from the fact that many students apply this without realizing it requires proof — the formula uses $X$ 's density, not $g(X)$ 's. The proof goes through a change of variables and verifies the formula gives the right answer.

LOTUS saves an enormous amount of effort. To compute $E[X^2]$ , you don't need to first find the distribution of $X^2$ ; just integrate $x^2 f_X(x)$ . To compute $E[\sin X]$ for a Normal $X$ , you don't need to find the distribution of $\sin X$ ; just integrate $\sin(x)$ against the Normal density.