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

Result R4788 on D4301: Conditional variance
Subresult of R3562: Real conditional variance partition into conditional moments

Formulation 0

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

(i)	$Y : \Omega \to \Xi$ is a D202: Random variable on $P$
(ii)	$X : \Omega \to \mathbb{R}$ is a D3161: Random real number on $P$
(iii)	\begin{equation} \mathbb{E} \|X\|^2 < \infty \end{equation}

Then \begin{equation} \text{Var}(X \mid Y) = \mathbb{E}(X^2 \mid Y) - (\mathbb{E}(X \mid Y))^2 \end{equation}

Proofs

Proof 0

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

(i)	$Y : \Omega \to \Xi$ is a D202: Random variable on $P$
(ii)	$X : \Omega \to \mathbb{R}$ is a D3161: Random real number on $P$
(iii)	\begin{equation} \mathbb{E} \|X\|^2 < \infty \end{equation}