# 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.

The validity of the material on this page is questionable.In particular: Every other definition of expectation I've seen uses a probability density function, not a cumulative density function.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Questionable}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Also known as

The **expectation** of a random variable $X$ is also called the **expected value of $X$** or the **mean value of $X$**.

For a given random variable, the **expectation** 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 mass function is a normalized weight function.

Various forms of $E$ can be seen to denote **expectation**:

- $\map E X$
- $\map {\mathrm E} X$
- $E \sqbrk X$
- $\mathop {\mathbb E} \sqbrk X$

and so on.

$\mathsf{Pr} \infty \mathsf{fWiki}$ uses $\expect X$ for notational consistency.

## 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.

## Technical Note

The $\LaTeX$ code for \(\expect {X}\) is `\expect {X}`

.

When the argument is a single character, it is usual to omit the braces:

`\expect X`

## Sources

There are no source works cited for this page.In particular: Would have to search for a source for thisSource citations are highly desirable, and mandatory for all definition pages.Definition pages whose content is wholly or partly unsourced are in danger of having such content deleted. To discuss this page in more detail, feel free to use the talk page. |