Definition:Expectation/Absolutely Continuous

Definition

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

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

Let $f_X$ be the probability density function of $X$.

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

$\ds \expect X = \int_{-\infty}^\infty x \map {f_X} x \rd x$

whenever:

$\ds \int_{-\infty}^\infty \size x \map {f_X} x \rd x < \infty$

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 Absolutely Continuous Random Variable shows that this definition is consistent with the general definition of expectation.

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

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): expectation (expected value)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): expectation (expected value)