Definition:Expectation/General Definition
Definition
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be an integrable real-valued random variable on $\struct {\Omega, \Sigma, \Pr}$.
The expectation of $X$, written $\expect X$, is defined by:
- $\ds \expect X = \int X \rd \Pr$
where the integral sign denotes the $\Pr$-integral of $X$.
This article, or a section of it, needs explaining. In particular: The link goes to $\mu$-integrable for a measure $\mu$, which itself is buried in the definition of a measure space. Worth recalling that $\Pr$ is itself the measure of a measure space (which is of course where measure spaces came from in the first place) You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. 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 {{Explain}} from the code. |
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.
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
- 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $10$: Probability: $10.1$: Basics