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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