Let $X$ be a \reftext{def:probability-space-random-variable-2026a}{random variable} on a probability space $(\Omega,\mathcal{F},P)$ with \reftext{def:distribution-cdf-random-variable-2026a}{distribution} $P_X$, let $\mathbb{R}$ denote the \reftext{def:real-numbers-c54-2026c}{real numbers}, and let $\varphi:\mathbb{R}\to\mathbb{R}$ be \reftext{def:measurable-function-2026a}{measurable} with respect to the \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel $\sigma$-algebra} on both sides. Then $\varphi\circ X$ is a random variable (preimages compose), and, with the \reftext{def:expectation-variance-2026a}{expectation} notation $\mathbb{E}$:

1. if $\varphi\ge 0$, then, with integrals as in \ref{def:lebesgue-integral-nonnegative-2026a},

2. in general, $\varphi\circ X$ is \reftext{def:lebesgue-integral-integrable-2026a}{integrable} with respect to $P$ if and only if $\varphi$ is integrable with respect to $P_X$, and in that case the displayed identity holds in $\mathbb{R}$.

In particular, \reftext{def:expectation-variance-2026a}{expectations, moments, and variances} of random variables depend only on their distributions, and identically distributed random variables share them.