The Rule of the Lazy Statistician

Suppose $X$ is a continuous random variable with PDF $f_X$, and we wish to compute the expected value of $Y = g(X)$ for some function $g.$ One option is to first derive the PDF of $Y$ and then integrate $y\, f_Y(y).$ The rule of the lazy statistician (LOTUS) lets us skip that step entirely.

If $X$ is a continuous random variable with PDF $f_X$ and $g$ is a real-valued function, then

$$E[g(X)] = \int_{-\infty}^{\infty} g(x)\, f_X(x)\, dx.$$

The name reflects the shortcut: we integrate $g$ directly against the PDF of $X,$ without ever finding the distribution of $g(X).$

For example, suppose $X$ is uniform on $[0,1],$ so $f_X(x) = 1$ for $x \in [0,1].$ Applying LOTUS with $g(x) = x^2{:}$

$$E[X^2] = \int_0^1 x^2 \cdot 1\, dx = \left[\dfrac{x^3}{3}\right]_0^1 = \dfrac{1}{3}.$$