Definition
Expectation
Let $X$ be a random variable on $(\Omega, \mathcal{F}, \mathbb{P})$ (Definition~Random Variable).
The expectation of $X$ is
\[
\mathbb{E}[X] := \int_\Omega X(\omega)\, d\mathbb{P}(\omega),
\]
provided the integral exists. For $p \geq 1$, we say $X \in L^p(\Omega, \mathcal{F}, \mathbb{P})$ if $\mathbb{E}[|X|^p] < \infty$.
Depends on
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