> [!NOTE] Theorem > Let $X$ be a [[Continuous random variables|continuous real-valued random variable]] whose [[Probability Density Function|probability density function]] is given by $f_{X}.$ Let $g:\mathbb{R}\to \mathbb{R}$ be a [[Real Function|real-valued function]]. Then $g(X)$ is [[Integrable Continuous Real-Valued Random Variable|integrable]] and its [[Expectation of Integrable Continuous Real-Valued Random Variable|expectation]] is given by $\mathbb{E}[X]=\int_{-\infty}^{\infty} g(X)\cdot f_{X}(x) \, dx $if the integral [[Absolutely Convergent Series|converges absolutely]]. **Proof**: First note that by [[Real-Valued Function of Continuous Real-Valued Random Variables]], $g(X)$ is indeed a real-valued random variable. ....