Definition:Expectation/Continuous

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

Let $F = \Pr \left({X < x}\right)$ be the cumulative probability function of $X$.

The expectation of $X$ is written $E \left({X}\right)$, and is defined over the probability measure as:


 * $\displaystyle E \left({X}\right) := \int_{x \mathop \in \Omega} x \ \mathrm d F$

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


 * $\displaystyle \int_{x \mathop \in \Omega} { \left\vert{x}\right\vert \ \mathrm d F } < \infty$

Also, from the definition of probability density function $f_X$ of $X$, we see it can also be written over the sample space:
 * $\displaystyle E \left({X}\right) := \int_{x \mathop \in \Omega_X} x \ f_X \left({x}\right) \ \mathrm d x$

Also known as
The expectation of $X$ is also called the expected value of $X$ or the mean of $X$, and (for a given continuous random variable) is often denoted $\mu$.

The terminology is appropriate, as it can be seen that an expectation is an example of a normalized weighted mean.

This follows from the fact that a probability density function is a normalized weight function.

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