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.
The validity of the material on this page is questionable. In particular: Every other definition of expectation I've seen uses a probability density function, not a cumulative density function. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues. 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 {{Questionable}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
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 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
Sources
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