User:Caliburn/s/prob/Expectation of Absolutely Continuous Random Variable

Theorem

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ be a absolutely continuous random variable on $\struct {\Omega, \Sigma, \Pr}$.

Let $f_X$ be a probability density function of $X$.

Then $X$ is integrable if and only if:

$\ds \int_{-\infty}^\infty \size x \map {f_X} x \rd x < \infty$

in which case:

$\ds \expect X = \int_{-\infty}^\infty x \map {f_X} x \rd x$