ThmDex – An index of mathematical definitions, results, and conjectures.

Conditional expectation of random real number conditioned on an independent random real number

Formulation 0

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that

(i)	$X, Y : \Omega \to \mathbb{R}$ are each a D3161: Random real number on $P$
(ii)	\begin{equation} \mathbb{E} \|X\|, \mathbb{E} \|Y\| < \infty \end{equation}
(iii)	$X, Y$ is an D2713: Independent random collection on $P$

Then \begin{equation} \mathbb{E}(X \mid Y) \overset{a.s.}{=} \mathbb{E}(X) \end{equation}

Formulation 1

Let $X, Y \in \text{Random}(\Omega \to \mathbb{R})$ each be a D3161: Random real number such that

(i)	\begin{equation} \mathbb{E} \|X\|, \mathbb{E} \|Y\| < \infty \end{equation}
(ii)	$X, Y$ is an D2713: Independent random collection

Then \begin{equation} \mathbb{E}(X \mid Y) \overset{a.s.}{=} \mathbb{E}(X) \end{equation}