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)