# 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 known as

The **expectation** of a random variable $X$ is also called the **expected value of $X$** or the **mean of $X$**, and (for a given 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 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.

## 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`

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