ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Deduction system
Zermelo-Fraenkel set theory
Binary cartesian set product
Binary relation
Simple map
Simple function
Measurable simple complex function
Simple integral
Unsigned basic integral
Unsigned basic expectation
Definition D5102
Basic expectation
Formulation 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $X : \Omega \to [-\infty, \infty]$ is a D4381: Random basic number on $P$
(ii) \begin{equation} \mathbb{E} |X| < \infty \end{equation}
The expectation of $X$ on $P$ is the D1699: Basic number \begin{equation} \mathbb{E}_{\mathbb{P}} X : = \mathbb{E}_{\mathbb{P}} X^+ - \mathbb{E}_{\mathbb{P}} X^- \end{equation}
Random real number moment
Countable indicator partition of complex expectation in terms of pullback events
Expectation of a random fraction need not equal fraction of expectations
Expectation of bounded random real number is within the bounding interval
Signed basic expectation of almost surely zero random number is zero