Definition:Expectation/Continuous

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

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

Let $F_X$ be the cumulative distribution function of $X$.

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


 * $\ds \expect X = \int_\R x \rd F_X$

whenever:


 * $\ds \int_\R \size x \rd F_X < \infty$

with the integrals being taken as Riemann-Stieltjes integrals.

Also see

 * Expectation of Continuous Random Variable as Riemann-Stieltjes Integral shows that this definition is consistent with the general definition of expectation.

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