Expectation of a function of a random variable

Compute the expectation of a transformed random variable using the distribution of the original.

Expectation of a function of a random variable

Law of the unconscious statistician: Let $X:\Omega\to S$ be a random variable with distribution (law) $\mu_X$ on $(S,\mathcal S)$ . If $g:S\to\mathbb R$ is measurable and $g(X)$ is integrable, then

\mathbb E[g(X)] = \int_\Omega g(X(\omega))\,d\mathbb P(\omega) = \int_S g(x)\,\mu_X(dx).

This identity lets you compute expectations by integrating against the distribution of $X$ rather than over the original probability space . When $\mu_X$ has a density with respect to Lebesgue measure (via the Radon–Nikodym theorem ), the right-hand side becomes an ordinary integral of $g(x)$ against that density.