Definition:Expectation/Absolutely Continuous

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

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

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

The expectation of $X$, written $\expect X$, is defined by:


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

whenever:


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

Also see

 * Expectation of Absolutely Continuous Random Variable shows that this definition is consistent with the general definition of expectation.