Definition:Expectation/Continuous

Definition
Let $X$ be a continuous random variable over the probability space $\struct {\Omega, \Sigma, \Pr}$.

Let $F = \map \Pr {X < x}$ be the cumulative probability function of $X$.

The expectation of $X$ is written $\expect X$, and is defined over the probability measure as:


 * $\expect X := \displaystyle \int_{x \mathop \in \Omega} x \rd F$

whenever the integral is absolutely convergent, i.e. when:


 * $\displaystyle \int_{x \mathop \in \Omega} \size x \rd F < \infty$

Also, from the definition of probability density function $f_X$ of $X$, we see it can also be written over the sample space:
 * $\displaystyle \expect X := \int_{x \mathop \in \Omega_X} x \, \map {f_X} x \rd x$

Also see
It can also be seen that the expectation of a continuous random variable is its first moment.