> [!NOTE] Definition 1 (Integrable Continuous Real-Valued Random Variable) > Let $X$ be a [[Continuous random variables|continuous real-valued random variable]] whose [[Probability Density Function|probability density function]] is given by $f_{X}.$ Then $X$ is integrable iff the integral $\int_{-\infty}^{\infty} x f_{X}(x) \, dx $[[Absolutely Convergent Series|converges absolutely]]: that is $\int_{-\infty}^{\infty}|x|f_{X}(x) \, dx<\infty.$ # Properties The [[Expectation of Integrable Continuous Real-Valued Random Variable|expectation]] of $X$ is given by $\mathbb{E}[X]= \int _{-\infty}^{\infty} x f_{X}(x) \, dx.$